††thanks: Corresponding author<EMAIL_ADDRESS>
# Heat statistics in the relaxation process of the Edwards-Wilkinson elastic
manifold

Yu-Xin Wu  School of Physics, Peking University, Beijing, 100871, China  Jin-
Fu Chen  School of Physics, Peking University, Beijing, 100871, China  Ji-Hui
Pei  School of Physics, Peking University, Beijing, 100871, China  Fan Zhang
School of Physics, Peking University, Beijing, 100871, China  H. T. Quan
School of Physics, Peking University, Beijing, 100871, China  Collaborative
Innovation Center of Quantum Matter, Beijing, 100871, China  Frontiers Science
Center for Nano-optoelectronics, Peking University, Beijing, 100871, China

(August 28, 2024)

###### Abstract

The stochastic thermodynamics of systems with a few degrees of freedom has
been studied extensively so far. We would like to extend the study to systems
with more degrees of freedom and even further–continuous fields with infinite
degrees of freedom. The simplest case for a continuous stochastic field is the
Edwards-Wilkinson elastic manifold. It is an exactly solvable model of which
the heat statistics in the relaxation process can be calculated analytically.
The cumulants require a cutoff spacing to avoid ultra-violet divergence. The
scaling behavior of the heat cumulants with time and the system size as well
as the large deviation rate function of the heat statistics in the large size
limit is obtained.

##  I Introduction

Historically, people studied thermodynamics in macroscopic systems such as
ideal gas with up to $10^{23}$ molecules. Due to the huge number of degrees of
freedom in the macroscopic scale, it is impossible to extract the trajectories
of individual particles explicitly. Hence it is not possible to study
thermodynamics of macroscopic systems in arbitrary far from equilibrium
processes. Nevertheless, for mesoscopic systems with only a few degrees of
freedom, stochastic dynamics (Langevin equation, Fokker-Planck equation,
master equation) provides detailed information about the system. Prominent
examples of mesoscopic systems include colloidal particles, macromolecules,
nanodevices and so on (Livi and Politi, 2017; Luca Peliti, 2021). In all these
examples, researchers focus on the dynamics of a few degrees of freedom of the
system while coarse-graining all the degrees of freedom of the reservoir.
Mesoscopic systems can be driven out of equilibrium by external driving, for
instance, by varying the temperature or by controlling them with optical
tweezers (Hummer and Szabo, 2001; Liphardt  _et al._ , 2002; Wang  _et al._ ,
2002; Blickle  _et al._ , 2006; Douarche  _et al._ , 2006; Harris  _et al._ ,
2007; Imparato  _et al._ , 2007; Toyabe  _et al._ , 2010; Gupta  _et al._ ,
2011; Alemany  _et al._ , 2012; Gieseler  _et al._ , 2014; Jun  _et al._ ,
2014; Koski  _et al._ , 2014; Lee  _et al._ , 2015; Martínez  _et al._ , 2015;
Hoang  _et al._ , 2018).

With the equation of motion, e.g., Langevin equation, Fokker-Planck equation
or master equation, researchers are able to establish a framework of
thermodynamics for mesoscopic systems in arbitrary far from equilibrium
processes. This is stochastic thermodynamics in which thermodynamic quantities
such as work, heat and entropy production in nonequilibrium processes have
been explored extensively in both classical and quantum realms (Jarzynski,
1997; Mazonka and Jarzynski, 1999; Narayan and Dhar, 2003; Speck and Seifert,
2004; van Zon and Cohen, 2004; Lua and Grosberg, 2005; Speck and Seifert,
2005; Taniguchi and Cohen, 2006; Imparato  _et al._ , 2007; Quan  _et al._ ,
2008; Engel, 2009; Fogedby and Imparato, 2009; Minh and Adib, 2009; Chatterjee
and Cherayil, 2010; Gomez-Solano  _et al._ , 2011; Nickelsen and Engel, 2011;
Speck, 2011; Kwon  _et al._ , 2013; Jiménez-Aquino and Velasco, 2013; Ryabov
_et al._ , 2013; Jarzynski  _et al._ , 2015; Salazar and Lira, 2016; Zhu  _et
al._ , 2016; Funo and Quan, 2018a, b; Hoang  _et al._ , 2018; Pagare and
Cherayil, 2019; Fogedby, 2020; Chen  _et al._ , 2021; Gupta and Sivak, 2021;
Chen and Quan, 2023; Paraguassú  _et al._ , 2023). In the study of work or
heat distribution for extreme nonequilibrium processes, rare events with
exponentially small probabilities have dominant contributions making finite
sampling error particularly serious. Hence previous studies, be it
experimental or computer simulations, are predominantly for small systems,
i.e., those with a few degrees of freedom (Hartmann, 2014). Nevertheless,
systems with a few degrees of freedom are too special. Therefore it is
desirable to extend the study of stochastic thermodynamics to more complicated
systems. We thus would like to extend the studies to systems with more degrees
of freedom, for example, stochastic fields. Hopefully in some exactly solvable
model we can obtain analytical results about work and heat distribution. These
rigorous results about work or heat distribution in systems with many degrees
of freedom not only have pedagogical value but also may bring some insights to
the understanding of thermodynamics in extreme nonequilibrium processes, as P.
W. Anderson once advocated, “More is different” (Anderson, 1972). While many
researchers are interested in the dynamic properties of stochastic fields
(Forrest and Tang, 1990; Antal and Rácz, 1996; Racz, 2003; Vvedensky, 2003;
Bustingorry  _et al._ , 2007), less research is carried out from the
perspective of stochastic thermodynamics except (Mallick  _et al._ , 2011; Wio
_et al._ , 2017; Rodríguez and Wio, 2019; Wio  _et al._ , 2020a, b) so far as
we know.

In this article we study the thermodynamics of an elastic manifold whose
underlying dynamics is described by the Edwards-Wilkinson (EW) equation
(Edwards and Wilkinson, 1982)

$\partial_{t}h(\boldsymbol{x},t)=\nu\nabla^{2}h(\boldsymbol{x},t)+\xi(\boldsymbol{x},t).$
(1)

where $h(\boldsymbol{x},t)$ is the local height at spatial point
$\boldsymbol{x}$ at time $t$, $\nu$ is the diffusive coefficient and
$\xi(\boldsymbol{x},t)$ is the Gaussian white noise.

The problem we analyze is the relaxation of an elastic manifold described by
the EW equation. The elastic manifold is initially put in contact with a heat
reservoir at the inverse temperature $\beta^{\prime}$. After initial
equilibration with the first heat reservoir at $\beta^{\prime}$ the system is
detached from it, and is put in contact with a second heat reservoir at the
inverse temperature $\beta$. The manifold subsequently tries to adapt to the
working temperature (Bustingorry  _et al._ , 2007). The relaxation is
characterized by the stochastic heat absorbed from/released into the
surrounding reservoir during a period of time $\tau$. We are interested in the
average and fluctuation of the heat in such a process. We find several generic
properties of the average and fluctuating heat in the relaxation process of
the EW elastic manifold. By employing the Feynman-Kac method (Chen  _et al._ ,
2021; Limmer  _et al._ , 2021), we obtain analytical results of the
characteristic function of heat for the EW model during an arbitrary
relaxation period $\tau$ with an arbitrary diffusive coefficient $\nu$ and
analyze the scaling behavior of the cumulants of heat with time. Analytical
results of the heat statistics bring important insights into understanding the
fluctuating property of heat in such a concrete and exactly solvable model. We
also verify from the analytical results that the heat statistics satisfy the
fluctuation theorem of heat exchange (Jarzynski and Wójcik, 2004). The large
deviation rate function of heat statistics in the large size limit is also
analyzed.

The rest of this article is organized as follows. In Section II we introduce
the EW model. In Section III we define the stochastic heat and obtain
analytical results of the characteristic function of heat using the Feynman-
Kac approach. We also compute the cumulants of heat and discuss their scaling
behavior with time and the system size. Conclusions are given in Section IV.

##  II The model

A $d$-dimensional elastic manifold, with finite size $2L$ in each direction,
joggles under thermal noise. Its local height $h(\boldsymbol{x},t)$ at spatial
point $\boldsymbol{x}$ at time $t$ evolves according to the EW equation Eq.
(1) which takes the form of a multivariable overdamped Langevin equation (Livi
and Politi, 2017). The thermal noise $\xi(\boldsymbol{x},t)$ is white in
nature, i.e., $\langle\xi(\boldsymbol{x},t)\rangle=0,$
$\langle\xi(\boldsymbol{x},t)\xi(\boldsymbol{x}^{\prime},t^{\prime})\rangle=\Gamma\delta(\boldsymbol{x}-\boldsymbol{x}^{\prime})\delta(t-t^{\prime}),$
with amplitude $\Gamma=2/\beta$. The EW energy is just that of a massless
field with Hamiltonian $H_{S}=\nu\int d\boldsymbol{x}(\nabla
h(\boldsymbol{x},t))^{2}/2$. Here the subscript $S$ refers to the system.

Initially, the system is prepared in an equilibrium state with the inverse
temperature $\beta^{\prime}$ characterized by a Gibbs-Boltzmann distribution
in the configuration space, i.e., the probability ${\cal P}(h,t)$ to find the
system in the configuration $\\{h(\boldsymbol{x},t)\\}$ is the Gibbs-Boltzmann
distribution

${\cal P}(h,0)={\cal
N}^{\prime-1}\exp\Big{[}-\beta^{\prime}\cdot\frac{\nu}{2}\int
d\boldsymbol{x}\Big{(}\nabla h(\boldsymbol{x},0)\Big{)}^{2}\Big{]}$ (2)

where ${\cal N}^{\prime}$ is the normalization constant

$\displaystyle{\cal N}^{\prime}$ $\displaystyle=\int
dh(\boldsymbol{x},0)\exp\Big{[}-\beta^{\prime}\cdot\frac{\nu}{2}\int
d\boldsymbol{x}\Big{(}\nabla h(\boldsymbol{x},0)\Big{)}^{2}\Big{]}.$ (3)

Here the integration in the normalization constant is taken over all possible
initial configurations while the one in the exponential factor is taken over
all spatial points.

After initial equilibration, the system is detached from the first heat
reservoir, and is placed in contact with a second heat reservoir at the
inverse temperature $\beta$, which is different from $\beta^{\prime}$. The
elastic manifold subsequently relaxes towards the equilibrium state at
temperature $\beta$ since no external driving is involved. The heat
absorbed/released is a fluctuating variable for the system undergoing
stochastic motion. We are interested in the heat statistics in such a
relaxation process.

For a finite-size manifold we take periodic boundary conditions along each
$\boldsymbol{x}$ direction. Following Refs. (Antal and Rácz, 1996; Livi and
Politi, 2017) we employ a Fourier representation of the height field

$h(\boldsymbol{x},t)=\frac{1}{(2\pi)^{d}}\underset{\boldsymbol{q}}{\sum}e^{i\boldsymbol{q}\cdot\boldsymbol{x}}h_{\boldsymbol{q}}(t),$
(4)

$h_{\boldsymbol{q}}(t)=\int
d\boldsymbol{x}e^{-i\boldsymbol{q}\cdot\boldsymbol{x}}h(\boldsymbol{x},t),$
(5)

where $\boldsymbol{q}$ represents a wavevector with $q_{j}=n_{j}\pi/L\
(j=x,y,z\dots,\ n_{j}=\pm 1,\pm 2...$ and
$h_{\boldsymbol{q}=\boldsymbol{0}}(t)=0$ for all time $t)$ (Bustingorry  _et
al._ , 2007).
The evolution of the Fourier component is given by

$\frac{\partial h_{\boldsymbol{q}}(t)}{\partial t}=-\nu
q^{2}h_{\boldsymbol{q}}(t)+\xi_{\boldsymbol{q}}(t),$ (6)

$\langle\xi_{\boldsymbol{q}}(t)\rangle=0,$ (7)

$\displaystyle\langle\xi_{\boldsymbol{q}}(t)\xi_{\boldsymbol{q^{\prime}}}(t^{\prime})\rangle$
$\displaystyle=\frac{2}{\beta}(2\pi)^{d}\delta(t-t^{\prime})\delta_{\boldsymbol{q},-\boldsymbol{q}^{\prime}}.$
(8)

The normalization constant in Eq. (3) can be computed as

$\displaystyle{\cal N}^{\prime}$ $\displaystyle=\int
d\\{h_{\boldsymbol{q}}(0)\\}\exp\Big{[}-\beta^{\prime}\nu\frac{1}{(2\pi)^{2d}}\underset{\boldsymbol{q}(q_{j}>0)}{\sum}q^{2}h_{\boldsymbol{q}}(0)h_{-\boldsymbol{q}}(0)\Big{]}$
$\displaystyle=\underset{\boldsymbol{q}(q_{j}>0)}{\prod}\frac{\pi(2\pi)^{2d}}{\beta^{\prime}\nu
q^{2}}.$ (9)

where $q^{2}$ stands for the modulus square of $\boldsymbol{q}.$

The probability density of the system state ${\cal P}(h,t)$ evolves under the
governing of the Fokker-Planck equation

$\displaystyle\frac{\partial{\cal P}(h,t)}{\partial t}$ $\displaystyle=-\int
d\boldsymbol{x}\frac{\delta}{\delta
h}\Big{[}\nu\nabla^{2}h(\boldsymbol{x},t){\cal P}(h,t)\Big{]}$
$\displaystyle\quad+\frac{\Gamma}{2}\int
d\boldsymbol{x}\frac{\delta^{2}}{\delta h^{2}}{\cal P}(h,t).$ (10)

In the Fourier space, the probability of the height field configuration is the
product of the real and the imaginary part over all modes

${\cal P}(\\{h_{\boldsymbol{q}}\\},t)=\underset{\boldsymbol{q}}{\prod}{\cal
P}(h_{\boldsymbol{q}},t)=\underset{\boldsymbol{q}}{\sum}{\cal
P}(h_{\boldsymbol{q}}^{R},t){\cal P}(h_{\boldsymbol{q}}^{I},t)$ (11)

where

$h_{\boldsymbol{q}}^{R}=\mathrm{Re}(h_{\boldsymbol{q}}),\quad
h_{\boldsymbol{q}}^{I}=\mathrm{Im}(h_{\boldsymbol{q}}).$ (12)

The Fokker-Planck equation in the Fourier space can be then written into two
independent parts: the real part and the imaginary part (Bettencourt, 2001)

$\displaystyle\frac{\partial{\cal P}(h_{\boldsymbol{q}}^{R,I},t)}{\partial t}$
$\displaystyle=\frac{(2\pi)^{d}}{2\beta}\frac{\partial^{2}{\cal
P}}{\partial(h_{\boldsymbol{q}}^{R,I})^{2}}+\nu q^{2}{\cal P}+\nu
q^{2}h_{\boldsymbol{q}}^{R,I}\frac{\partial{\cal P}}{\partial
h_{\boldsymbol{q}}^{R,I}}.$ (13)

Having introduced the model, in the following we will calculate the heat
statistics in the relaxation process.

##  III Heat Statistics

In this section we study heat statistics of the EW elastic manifold in the
relaxation process. First, we obtain the analytical results of heat statistics
and verify the fluctuation theorem of heat exchange. Second, we study the
asymptotic behavior of the cumulants. Third, we calculate the large deviation
function of heat statistics in the large size limit.

###  III.1 Characteristic function

Since no external driving is applied to the system, no work is performed
during the relaxation process. The fluctuating heat $Q$ absorbed from the heat
reservoir equals the energy difference between the initial and the final
states over a time period $\tau$

$Q=H_{S}(h(x,\tau))-H_{S}(h(x,0)).$ (14)

The characteristic function of heat $\chi_{\tau}(u)$ is defined as the Fourier
transform of the heat distribution

$\chi_{\tau}(u)=\int dQ\exp(iuQ){\cal P}(Q,\tau).$ (15)

Here ${\cal P}(Q,\tau)$ stands for the probability of the heat $Q$ transferred
from the heat reservoir to the system during the period of time $\tau$. The
characteristic function of heat $\chi_{\tau}(u)$ can be calculated using the
Feynman-Kac approach (Chen  _et al._ , 2021; Limmer  _et al._ , 2021; Chen and
Quan, 2023)

$\displaystyle\chi_{\tau}(u)$ $\displaystyle=\langle\exp(iuQ)\rangle$
$\displaystyle=\int dhe^{iuH_{S}(h(x,\tau))}\eta(h,\tau)$ (16)

where the probability-density-like function $\eta(h,\tau)$ satisfies Eq. (10)
and Eq. (13) with the initial condition

$\displaystyle\eta(h,0)$ $\displaystyle=e^{-iuH_{S}(h(x,0))}{\cal P}(h,0).$
(17)

The probability-density-like function $\eta(h,\tau)$ is solved in the Fourier
space (See Appendix A for detailed derivation) and we obtain the
characteristic function of heat for the relaxation process over a time period
of $\tau$

$\chi_{\tau}(u)=\beta\beta^{\prime}\underset{\boldsymbol{q}(q_{j}\geq\frac{\pi}{L})}{\prod}\frac{\exp(2\nu
q^{2}\tau)}{-u(i\beta^{\prime}-i\beta-u)\Big{[}\exp(2\nu
q^{2}\tau)-1\Big{]}+\beta\beta^{\prime}\exp(2\nu q^{2}\tau)}.$ (18)

The wavevector component in each direction only takes positive discrete values
$q_{j}=n_{j}\pi/L,n_{j}=1,2...$

We do the self-consistent check of the analytic result Eq. (18) from three
aspects:
1\. The distribution of heat satisfies the conservation of probability

$\chi_{\tau}(0)=1.$ (19)

2\. One can see the characteristic function of heat exhibits the following
symmetry:

$\chi_{\tau}(u)=\chi_{\tau}(i\beta^{\prime}-i\beta-u),$ (20)

indicating that the heat distribution satisfies the fluctuation theorem of
heat exchange (Jarzynski and Wójcik, 2004; van Zon and Cohen, 2004; Chen and
Quan, 2023)

$\displaystyle\langle e^{iuQ}\rangle$ $\displaystyle=\langle
e^{(-iu+\beta-\beta^{\prime})Q}\rangle.$ (21)

By setting $u=0$, we obtain the relation
$\chi_{\tau}(i\beta^{\prime}-i\beta)=1$, which is exactly the fluctuation
theorem of heat exchange in the integral form
$\langle\exp[-(\beta^{\prime}-\beta)Q]\rangle=1$(Jarzynski and Wójcik, 2004).
3\. In the long time limit $\tau\rightarrow\infty$, the characteristic
function becomes

$\displaystyle\underset{\tau\to\infty}{\lim}\chi_{\tau}(u)$
$\displaystyle=\underset{\boldsymbol{q}(q_{j}\geq\frac{\pi}{L})}{\prod}\frac{\beta\beta^{\prime}}{(u+i\beta)(u-i\beta^{\prime})}.$

This result, independent of the relaxation dynamics, can be written in the
form

$\displaystyle\underset{\tau\to\infty}{\lim}\chi_{\tau}(u)$
$\displaystyle=\Big{\langle}e^{iuH_{S}(h(x,\tau))}\Big{\rangle}_{\beta}\Big{\langle}e^{-iuH_{S}(h(x,0))}\Big{\rangle}_{\beta^{\prime}}$
(22)

where the initial distribution (thermal equilibrium with the inverse
temperature $\beta^{\prime}$) and the final distribution (thermal equilibrium
with the inverse temperature $\beta$) are sampled independently, reflecting
the complete thermalization of the system (Fogedby and Imparato, 2009). This
result agrees with our intuition.

(a)

(b)

Figure 1: Average heat as a function of time. Parameters for both panels:
$d=1,\ \nu=1,\ \beta^{\prime}=4,\ \beta=2.$ (a) $\langle Q\rangle$ as a
function of $\tau$ for three system sizes $L=30,35,40$, fixing $a=0.2.$ Inset:
the saturation value of average heat $\langle Q\rangle_{st}$ as a function of
system size $L$. (b) $\langle Q\rangle$ as a function of $\tau$ for three
cutoff spacings $a=0.2,0.5,1.0$, fixing $L=10.$ Inset: the saturation value of
average heat $\langle Q\rangle_{st}$ as a function of cutoff spacing $a$.

###  III.2 Cumulants

The cumulants of heat can be derived by taking derivatives of the logarithm of
the characteristic function $\chi_{\tau}(u)$ with respect to $u$ at $u=0$,
with the first cumulant representing the average heat and the second one
standing for the variance.

The average heat is

$\displaystyle\langle Q\rangle$
$\displaystyle=\frac{1}{i}\frac{d\ln\chi_{\tau}(u)}{du}|_{u=0}$
$\displaystyle=\underset{\boldsymbol{q}(\frac{\pi}{a}\geq
q_{j}\geq\frac{\pi}{L})}{\sum}\frac{\Big{[}1-\exp(-2\nu
q^{2}\tau)\Big{]}(\beta^{\prime}-\beta)}{\beta\beta^{\prime}}$
$\displaystyle=\frac{\beta^{\prime}-\beta}{\beta\beta^{\prime}}\Big{(}\frac{\pi}{L}\Big{)}^{-d}\int_{\frac{\pi}{L}}^{\frac{\pi}{a}}d\boldsymbol{q}\Big{[}1-\exp(-2\nu
q^{2}\tau)\Big{]}.$ (23)

A cutoff $\pi/a$ of the wavevector is needed to avoid ultra-violet divergence,
i.e., we introduce a smallest spacing $a$ in this elastic manifold (Kerson
Huang, 1987; Parisi and Machta, 1989; Livi and Politi, 2017). Since we
consider a continuous field, the cutoff spacing is always much smaller than
the system size $a\ll L$. We will see that the choice of the value of $a$ will
influence the average heat (See Fig. 1 (b) inset plot).

Figure 2: Average heat of a mode $\boldsymbol{q}$ for different time
durations. The parameters take values $L=30,\ d=1,\ \nu=1,\ \beta^{\prime}=4,\
\beta=2$ and the curves correspond to three values of time delay
$\tau=10^{1},10^{0},10^{-1}$ from the bottom to the top. The dashed line
stands for the saturation value.

We rewrite the average heat $\langle Q\rangle$ with a change of the variable
$\boldsymbol{s}=L\boldsymbol{q}$

$\displaystyle\langle Q\rangle$
$\displaystyle=\frac{(\beta^{\prime}-\beta)}{\beta\beta^{\prime}\pi{}^{d}}f\Big{(}\frac{\nu\tau}{L^{2}}\Big{)},$
(24)

where

$\displaystyle f(r)$
$\displaystyle=\int_{\pi}^{\frac{L\pi}{a}}d\boldsymbol{s}\Big{[}1-e^{-2rs^{2}}\Big{]}$
$\displaystyle=(\frac{L-a}{a}\pi)^{d}+(\frac{\pi}{8r})^{\frac{d}{2s}}\Big{[}\mathrm{Erf}(\pi\sqrt{2r})-\mathrm{Erf}(\frac{\pi
L\sqrt{2r}}{a})\Big{]}^{d}.$ (25)

$\mathrm{Erf}(r)$ is the error function.

In the following we discuss the asymptotic behavior of the average heat as a
function of time. For one-dimensional case, the average heat as a function of
time is illustrated in Fig. 1. At the initial stage, for $\tau\ll a^{2}/\nu$,

$\displaystyle\langle Q\rangle$
$\displaystyle\approx\frac{2\pi^{2}}{3a^{2}}\frac{(\beta^{\prime}-\beta)}{\beta\beta^{\prime}}\nu\tau\frac{L}{a}.$
(26)

The average heat initially increases with time linearly. This is Newton’s law
of cooling.

For the intermediate time $a^{2}/\nu\ll\tau\ll L^{2}/\nu$,

$\displaystyle\langle Q\rangle$
$\displaystyle\approx\frac{(\beta^{\prime}-\beta)}{\beta\beta^{\prime}}\frac{L}{a}\Big{(}1-\frac{a}{\sqrt{8\nu}}\tau^{-1/2}\Big{)}.$
(27)

It exhibits $\tau^{-1/2}$ scaling with time.

In the long time limit, for $\tau\gg L^{2}/\nu$,

$\displaystyle\langle Q\rangle$
$\displaystyle\rightarrow\frac{\beta^{\prime}-\beta}{\beta\beta^{\prime}}\frac{L}{a},$
(28)

the average heat saturates, which is a consequence of the equipartition
theorem. The saturation value of heat is an extensive quantity which scales
linearly with the system size $L$. It will not diverge for a finite spacing
$a$ as a result of finite resolution.

From Eq. (23) one can see the average heat for every $\boldsymbol{q}$ mode is

$\langle
Q_{\boldsymbol{q}}\rangle=\frac{\beta^{\prime}-\beta}{\beta\beta^{\prime}}\Big{(}\frac{\pi}{L}\Big{)}^{-d}\Big{[}1-\exp(-2\nu
q^{2}\tau)\Big{]}.$ (29)

As we can see from this equation and Fig. 2, heat transfer occurs mainly
through high-energy modes and occurs in high-energy modes more quickly than
that in lower ones.

For fixed time duration $\tau$, in the small wavevector limit, i.e., $2\nu
q^{2}\tau\ll 1$, it increases with time linearly

$\langle
Q_{\boldsymbol{q}}\rangle=2\nu\tau\frac{\beta^{\prime}-\beta}{\beta\beta^{\prime}}\Big{(}\frac{\pi}{L}\Big{)}^{-d}q^{2},$
(30)

which is the Newton’s law of cooling.

On the other hand, if one takes the large wavevector limit, i.e., $2\nu
q^{2}\tau\gg 1$, the average heat reaches the asymptotic value

$\langle
Q_{\boldsymbol{q}}\rangle=\frac{\beta^{\prime}-\beta}{\beta\beta^{\prime}}\Big{(}\frac{\pi}{L}\Big{)}^{-d},$
(31)

which is the result of the equipartition theorem.

(a)

(b)

Figure 3: Variance of heat as a function of time. Parameters for both panels:
$d=1,\ \nu=1,\ \beta^{\prime}=4,\ \beta=2.$ (a) $\mathrm{var}(Q)$ as a
function of $\tau$ for three system sizes $L=30,35,40$, fixing $a=0.2.$ Inset:
the saturation value of heat variance $\mathrm{var}(Q)_{st}$ as a function of
system size $L$. (b) $\mathrm{var}(Q)$ as a function of $\tau$ for three
cutoff spacings $a=0.2,0.25,0.3$, fixing $L=10.$ Inset: the saturation value
of heat variance $\mathrm{var}(Q)_{st}$ as a function of cutoff spacing $a$.

From the analytical result of heat statistics Eq. (18) we can also study the
variance of heat. The variance of heat is defined as $\mathrm{var}(Q)=\langle
Q^{2}\rangle-\langle Q\rangle^{2}$, and can be calculated as

$\displaystyle\mathrm{var}(Q)$
$\displaystyle=\frac{1}{i^{2}}\frac{d^{2}\ln\chi_{\tau}(u)}{du^{2}}|_{u=0}$
$\displaystyle=\Big{(}\frac{\pi}{L}\Big{)}^{-d}\frac{1}{\beta^{2}\beta^{\prime
2}}\int_{\frac{\pi}{L}}^{\frac{\pi}{a}}d\boldsymbol{q}e^{-4\nu
q^{2}\tau}(-1+e^{2\nu q^{2}\tau})$ $\displaystyle\quad\bigg{[}(-1+e^{2\nu
q^{2}\tau})\beta^{2}+2\beta\beta^{\prime}+(-1+e^{2\nu q^{2}\tau})\beta^{\prime
2}\bigg{]}$ $\displaystyle=\frac{1}{\beta^{2}\beta^{\prime
2}\pi^{d}}g\Big{(}\frac{\nu\tau}{L^{2}}\Big{)}$ (32)

where

$\displaystyle g(r)$
$\displaystyle=\int_{\pi}^{\frac{L\pi}{a}}d\boldsymbol{s}\Big{[}(\beta^{2}+\beta^{\prime
2})(1-2e^{-2rs^{2}}+e^{-4rs^{2}})$
$\displaystyle\quad+2\beta\beta^{\prime}(-e^{-4rs^{2}}+e^{-2rs^{2}})\Big{]}.$

In the one-dimensional case, for $\tau\ll a^{2}/\nu$, we have

$\mathrm{var}(Q,\tau)\approx\frac{4\pi^{2}}{3a^{2}\beta\beta^{\prime}}\nu\tau\frac{L}{a}.$
(33)

It grows with time linearly in the very beginning.

For $a^{2}/\nu\ll\tau\ll L^{2}/\nu$,

$\displaystyle\mathrm{var}(Q,\tau)$
$\displaystyle\approx\frac{4\pi^{4}\nu^{2}\tau^{2}}{5\beta^{2}\beta^{\prime
2}a^{4}}(\beta^{2}-3\beta\beta^{\prime}+\beta^{\prime 2})\frac{L}{a}.$ (34)

It scales as $\tau^{2}$ as time elapses.

Finally, for $\tau\gg L^{2}/\nu$, it reaches the saturation value in the long
time,

$\displaystyle\mathrm{var}(Q,\tau)$
$\displaystyle\approx\frac{\beta^{2}+\beta^{\prime 2}}{\beta^{2}\beta^{\prime
2}}\frac{L}{a}.$ (35)

As can be seen from Fig. 3, the variance of heat depends on the cutoff spacing
$a$ as well. Similar to the average heat, the saturation value of variance
increases linearly with the system size $L$ and will not diverge for finite
spacing $a$. Higher order cumulants of heat can be analyzed in a similar way.

###  III.3 Large deviation rate function

We can also study the large deviation rate function of the heat statistics in
the large size limit.

The scaled cumulant generating function (SCGF) $\phi(u,\tau)$ of heat per
volume over time $\tau$, which is defined through

$\langle\exp[(2L)^{d}u\frac{Q}{(2L)^{d}}]\rangle\asymp_{L\to\infty}e^{(2L)^{d}\phi(u,\tau)}$
(36)

or

$\displaystyle\phi(u,\tau)$
$\displaystyle=\underset{L\to\infty}{\lim}\frac{1}{(2L)^{d}}\ln\langle\exp[(2L)^{d}u\frac{Q}{(2L)^{d}}]\rangle$
$\displaystyle=\underset{L\to\infty}{\lim}\frac{1}{(2L)^{d}}\ln\chi_{\tau}(-iu),$
(37)

can be computed by

$\displaystyle\phi(u,\tau)$
$\displaystyle=\underset{L\to\infty}{\lim}-\frac{1}{(2\pi)^{d}}\int_{\frac{\pi}{L}}^{\frac{\pi}{a}}d\boldsymbol{q}\ln\Big{(}\frac{-u(\beta^{\prime}-\beta+u)}{\beta\beta^{\prime}}\Big{[}1-\exp(-2\nu
q^{2}\tau)\Big{]}+1\Big{)}.$

The large deviation rate function for heat per volume over time $\tau$ is just
the Legendre-Fenchel transform of the SCGF (Touchette, 2009)

$\displaystyle I(\frac{Q}{(2L)^{d}},\tau)$
$\displaystyle=\underset{L\to\infty}{\lim}-\frac{1}{(2L)^{d}}\ln{\cal
P}(\frac{Q}{(2L)^{d}},\tau)$ $\displaystyle=\underset{u\mathbb{\in
R}}{\sup}\Big{\\{}u\frac{Q}{(2L)^{d}}-\phi(u,\tau)\Big{\\}}.$ (38)

We emphasize that the large deviation rate function of work distribution in
the large size limit has been studied in other models previously (See e.g.,
Refs. (Gambassi and Silva, 2012; Hartmann, 2014)). But as far as we know, the
large deviation function of heat in the large size limit has not been reported
previously.

With the large deviation rate function Eq. (38), we can write down the
probability distribution of heat per volume over time $\tau$ as

${\cal
P}(\frac{Q}{(2L)^{d}},\tau)\asymp_{L\to\infty}\exp\Big{[}-(2L)^{d}I(\frac{Q}{(2L)^{d}},\tau)\Big{]},$
(39)

which demonstrates the dependence of the heat distribution on the system size.
And the fluctuation theorem of heat exchange Eq. (21) can also be formulated
in terms of the large deviation rate function.

##  IV Conclusion

Previously, the stochastic thermodynamics of systems with a few degrees of
freedom have been studied extensively both in classical and quantum realms
(Jarzynski, 1997; Mazonka and Jarzynski, 1999; Narayan and Dhar, 2003; Speck
and Seifert, 2004; van Zon and Cohen, 2004; Lua and Grosberg, 2005; Speck and
Seifert, 2005; Taniguchi and Cohen, 2006; Imparato  _et al._ , 2007; Quan  _et
al._ , 2008; Engel, 2009; Fogedby and Imparato, 2009; Minh and Adib, 2009;
Chatterjee and Cherayil, 2010; Gomez-Solano  _et al._ , 2011; Nickelsen and
Engel, 2011; Speck, 2011; Kwon  _et al._ , 2013; Jiménez-Aquino and Velasco,
2013; Ryabov  _et al._ , 2013; Jarzynski  _et al._ , 2015; Salazar and Lira,
2016; Zhu  _et al._ , 2016; Funo and Quan, 2018a, b; Hoang  _et al._ , 2018;
Pagare and Cherayil, 2019; Fogedby, 2020; Chen  _et al._ , 2021; Gupta and
Sivak, 2021; Chen and Quan, 2023; Paraguassú  _et al._ , 2023). However, less
is known in systems with many degrees of freedom. What new results the
complexity of many degrees of freedom will bring to stochastic thermodynamics
remains largely unexplored.

In this article, we extend previous studies about the stochastic
thermodynamics of systems with a few degrees of freedom to a continuous field.
We compute the heat statistics in the relaxation process of an exactly
solvable model — an elastic manifold whose underlying dynamics can be
described by the Edwards-Wilkinson equation. By employing Feynman-Kac
approach, we calculate analytically the characteristic function of heat for
any relaxation time. The analytical results of heat statistics have
pedagogical value and may bring important insights to the understanding of
thermodynamics in extreme nonequilibrium processes. For example, the cumulants
of heat in such a system with many degrees of freedom require a spatial cutoff
to avoid the ultra-violet divergence, which is a consequence of finite
resolution. We also analyze the scaling behavior of the cumulants with time
and the system size. In addition, the large deviation rate function of heat in
the large size limit is analyzed.

This work can be regarded as an early step in the stochastic thermodynamics of
continuous fields. More interesting problems remain to be explored such as the
definitions for the thermodynamic quantities in every space-time point, the
extension to nonlinear models, the work statistics in the presence of external
driving and so on. Studies about these issues will be given in our future
work.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(NSFC) under Grants No. 12147157, No. 11775001, and No. 11825501.

##  Appendix A Derivation of Eq. (50)

Similar to the probability density distribution, the modified function
$\eta(h,t)$ can be written as the product of the imaginary part and the real
part over all modes in the Fourier space

${\cal\eta}(\\{h_{\boldsymbol{q}}\\},t)=\underset{q_{i}\geq\pi/L}{\prod}{\cal\eta}_{\boldsymbol{q}}(h_{\boldsymbol{q}}^{R},t){\cal\eta}_{\boldsymbol{q}}(h_{\boldsymbol{q}}^{I},t).$
(40)

The probability-density-like function $\eta(h,t)$ follows the same time
evolution as ${\cal P}(h,t)$ in Eq. (13)

$\displaystyle\frac{\partial{\cal\eta}_{\boldsymbol{q}}(h_{\boldsymbol{q}}^{R,I},t)}{\partial
t}$
$\displaystyle=\frac{(2\pi)^{d}}{2\beta}\frac{\partial^{2}{\cal\eta}_{\boldsymbol{q}}}{\partial(h_{\boldsymbol{q}}^{R,I})^{2}}+\nu
q^{2}{\cal\eta}_{\boldsymbol{q}}+\nu
q^{2}h_{\boldsymbol{q}}^{R,I}\frac{\partial{\cal\eta}_{\boldsymbol{q}}}{\partial
h_{\boldsymbol{q}}^{R,I}}.$ (41)

with the initial condition

$\eta(h,0)=e^{-iuH_{S}(0)}{\cal P}(h,0).$ (42)

Due to the quadratic nature of the EW equation, we assume the time-dependent
solution $\eta(h,t)$ takes a Gaussian form at any time

$\displaystyle{\cal\eta}_{\boldsymbol{q}}(h_{\boldsymbol{q}}^{R,I},t)$
$\displaystyle=\sqrt{\frac{\beta^{\prime}\nu
q^{2}}{\pi(2\pi)^{2d}}}\exp\Big{[}-A(t)(h_{\boldsymbol{q}}^{R,I})^{2}+B(t)\Big{]}.$
(43)

The coefficients are governed by the following ordinary differential equations

$\dot{A}(t)=-\frac{2(2\pi)^{d}}{\beta}A^{2}(t)+2A(t)\nu q^{2},$ (44)

$\dot{B}(t)=-\frac{(2\pi)^{d}}{\beta}A(t)+\nu q^{2}.$ (45)

The initial condition Eq. (42) gives way to the initial values of the
coefficients

$A(0)=(\beta^{\prime}+iu)\nu\frac{1}{(2\pi)^{d}}q^{2},$ (46)

$B(0)=0.$ (47)

By solving the above equations we obtain

$\displaystyle A(t)$ $\displaystyle=\frac{1}{(2\pi)^{d}}\frac{e^{2\nu
q^{2}t}\beta(u-i\beta^{\prime})\nu q^{2}}{(e^{2\nu
q^{2}t}-1)u-i[\beta+(e^{2\nu q^{2}t}-1)\beta^{\prime}]},$ (48) $\displaystyle
B(t)$ $\displaystyle=\nu
q^{2}t+\frac{1}{2}\ln\left[\frac{i\beta}{u-i\beta^{\prime}+i\beta+(i\beta^{\prime}-u)e^{2\nu
q^{2}t}}\right].$ (49)

Substituting Eqs. (48) and (49) into Eq. (43), we arrive at

$\displaystyle{\cal\eta}(\\{h_{\boldsymbol{q}}\\},t)$
$\displaystyle=\underset{q_{i}\geq\pi/L}{\prod}{\cal\eta}_{\boldsymbol{q}}(h_{\boldsymbol{q}}^{R},t){\cal\eta}_{\boldsymbol{q}}(h_{\boldsymbol{q}}^{I},t)$
$\displaystyle=\underset{\boldsymbol{q}(q_{i}\geq\pi/L)}{\prod}\frac{\beta^{\prime}\nu
q^{2}}{\pi(2\pi)^{d}}\frac{i\beta\exp(2\nu
q^{2}\tau)}{u-i\beta^{\prime}+i\beta+(i\beta^{\prime}-u)\exp(2\nu q^{2}t)}$
$\displaystyle\hfill\exp\bigg{\\{}-\frac{1}{(2\pi)^{d}}\frac{\exp(2\nu
q^{2}t)\beta(u-i\beta^{\prime})\nu q^{2}}{\Big{[}-1+\exp(2\nu
q^{2}t)\Big{]}u-i\Big{[}\beta-\beta^{\prime}+\beta^{\prime}\exp(2\nu
q^{2}t)\Big{]}}\Big{[}(h_{\boldsymbol{q}}^{R})^{2}+(h_{\boldsymbol{q}}^{I})^{2}\Big{]}\bigg{\\}}.$
(50)

Substituting it into Eq. (16), we obtain the characteristic function of heat
Eq. (18) of the EW elastic manifold in the relaxation process.

## References

  * Livi and Politi (2017) R. Livi and P. Politi,  _Nonequilibrium Statistical Physics_ (Cambridge University Press, 2017).
  * Luca Peliti (2021) S. P. Luca Peliti,  _Stochastic Thermodynamics_ (Princeton University Press, 2021).
  * Hummer and Szabo (2001) G. Hummer and A. Szabo, Proc. Natl. Acad. Sci. 98, 3658 (2001).
  * Liphardt  _et al._ (2002) J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, and C. Bustamante, Science 296, 1832 (2002).
  * Wang  _et al._ (2002) G. M. Wang, E. M. Sevick, E. Mittag, D. J. Searles, and D. J. Evans, Phys. Rev. Lett. 89, 050601 (2002).
  * Blickle  _et al._ (2006) V. Blickle, T. Speck, L. Helden, U. Seifert, and C. Bechinger, Phys. Rev. Lett. 96, 070603 (2006).
  * Douarche  _et al._ (2006) F. Douarche, S. Joubaud, N. B. Garnier, A. Petrosyan, and S. Ciliberto, Phys. Rev. Lett. 97, 140603 (2006).
  * Harris  _et al._ (2007) N. C. Harris, Y. Song, and C.-H. Kiang, Phys. Rev. Lett. 99, 068101 (2007).
  * Imparato  _et al._ (2007) A. Imparato, L. Peliti, G. Pesce, G. Rusciano, and A. Sasso, Phys. Rev. E 76, 050101 (2007).
  * Toyabe  _et al._ (2010) S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Nat. Phys. 6, 988 (2010).
  * Gupta  _et al._ (2011) A. N. Gupta, A. Vincent, K. Neupane, H. Yu, F. Wang, and M. T. Woodside, Nat. Phys. 7, 631 (2011).
  * Alemany  _et al._ (2012) A. Alemany, A. Mossa, I. Junier, and F. Ritort, Nat. Phys. 8, 688 (2012).
  * Gieseler  _et al._ (2014) J. Gieseler, R. Quidant, C. Dellago, and L. Novotny, Nat. Nanotechnology 9, 358 (2014).
  * Jun  _et al._ (2014) Y. Jun, M. Gavrilov, and J. Bechhoefer, Phys. Rev. Lett. 113, 190601 (2014).
  * Koski  _et al._ (2014) J. Koski, V. Maisi, T. Sagawa, and J. Pekola, Phys. Rev. Lett. 113, 030601 (2014).
  * Lee  _et al._ (2015) D. Y. Lee, C. Kwon, and H. K. Pak, Phys. Rev. Lett. 114, 060603 (2015).
  * Martínez  _et al._ (2015) I. A. Martínez, É. Roldán, L. Dinis, D. Petrov, J. M. R. Parrondo, and R. A. Rica, Nat. Phys. 12, 67 (2015).
  * Hoang  _et al._ (2018) T. M. Hoang, R. Pan, J. Ahn, J. Bang, H. Quan, and T. Li, Phys. Rev. Lett. 120, 080602 (2018).
  * Jarzynski (1997) C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).
  * Mazonka and Jarzynski (1999) O. Mazonka and C. Jarzynski, (1999), arXiv:cond-mat/9912121 [cond-mat.stat-mech] .
  * Narayan and Dhar (2003) O. Narayan and A. Dhar, J. Phys. A: Math. Gen. 37, 63 (2003).
  * Speck and Seifert (2004) T. Speck and U. Seifert, Phys. Rev. E 70, 066112 (2004).
  * van Zon and Cohen (2004) R. van Zon and E. G. D. Cohen, Phys. Rev. E 69, 056121 (2004).
  * Lua and Grosberg (2005) R. C. Lua and A. Y. Grosberg, The Journal of Physical Chemistry B 109, 6805 (2005).
  * Speck and Seifert (2005) T. Speck and U. Seifert, Eur. Phys. J. B 43, 521 (2005).
  * Taniguchi and Cohen (2006) T. Taniguchi and E. G. D. Cohen, J. Stat. Phys. 126, 1 (2006).
  * Quan  _et al._ (2008) H. T. Quan, S. Yang, and C. P. Sun, Phys. Rev. E 78, 021116 (2008).
  * Engel (2009) A. Engel, Phys. Rev. E 80, 021120 (2009).
  * Fogedby and Imparato (2009) H. C. Fogedby and A. Imparato, J. Phys. A: Math. Theor. 42, 475004 (2009).
  * Minh and Adib (2009) D. D. L. Minh and A. B. Adib, Phys. Rev. E 79, 021122 (2009).
  * Chatterjee and Cherayil (2010) D. Chatterjee and B. J. Cherayil, Phys. Rev. E 82, 051104 (2010).
  * Gomez-Solano  _et al._ (2011) J. R. Gomez-Solano, A. Petrosyan, and S. Ciliberto, Phys. Rev. Lett. 106, 200602 (2011).
  * Nickelsen and Engel (2011) D. Nickelsen and A. Engel, Eur. Phys. J. B 82, 207 (2011).
  * Speck (2011) T. Speck, J. Phys. A: Math. Theor. 44, 305001 (2011).
  * Kwon  _et al._ (2013) C. Kwon, J. D. Noh, and H. Park, Phys. Rev. E 88, 062102 (2013).
  * Jiménez-Aquino and Velasco (2013) J. I. Jiménez-Aquino and R. M. Velasco, Phys. Rev. E 87, 022112 (2013).
  * Ryabov  _et al._ (2013) A. Ryabov, M. Dierl, P. Chvosta, M. Einax, and P. Maass, J. Phys. A: Math. Theor. 46, 075002 (2013).
  * Jarzynski  _et al._ (2015) C. Jarzynski, H. Quan, and S. Rahav, Phys. Rev. X 5, 031038 (2015).
  * Salazar and Lira (2016) D. S. P. Salazar and S. A. Lira, J. Phys. A: Math. Theor. 49, 465001 (2016).
  * Zhu  _et al._ (2016) L. Zhu, Z. Gong, B. Wu, and H. T. Quan, Phys. Rev. E 93, 062108 (2016).
  * Funo and Quan (2018a) K. Funo and H. Quan, Phys. Rev. Lett. 121, 040602 (2018a).
  * Funo and Quan (2018b) K. Funo and H. T. Quan, Phys. Rev. E 98, 012113 (2018b).
  * Pagare and Cherayil (2019) A. Pagare and B. J. Cherayil, Phys. Rev. E 100, 052124 (2019).
  * Fogedby (2020) H. C. Fogedby, J. Stat. Mech.: Theory Exp. 2020, 083208 (2020).
  * Chen  _et al._ (2021) J.-F. Chen, T. Qiu, and H.-T. Quan, Entropy 23, 1602 (2021).
  * Gupta and Sivak (2021) D. Gupta and D. A. Sivak, Phys. Rev. E 104, 024605 (2021).
  * Chen and Quan (2023) J.-F. Chen and H. T. Quan, Phys. Rev. E 107, 024135 (2023).
  * Paraguassú  _et al._ (2023) P. V. Paraguassú, R. Aquino, and W. A. Morgado, Physica A 615, 128568 (2023).
  * Hartmann (2014) A. K. Hartmann, Phys. Rev. E 89, 052103 (2014).
  * Anderson (1972) P. W. Anderson, Science 177, 393 (1972).
  * Forrest and Tang (1990) B. M. Forrest and L.-H. Tang, Phys. Rev. Lett. 64, 1405 (1990).
  * Antal and Rácz (1996) T. Antal and Z. Rácz, Phys. Rev. E 54, 2256 (1996).
  * Racz (2003) Z. A. Racz, in  _SPIE Proceedings_, edited by M. B. Weissman, N. E. Israeloff, and A. S. Kogan (SPIE, 2003).
  * Vvedensky (2003) D. D. Vvedensky, Phys. Rev. E 67, 025102 (2003).
  * Bustingorry  _et al._ (2007) S. Bustingorry, L. F. Cugliandolo, and J. L. Iguain, J. Stat. Mech.: Theory Exp. 2007, P09008 (2007).
  * Mallick  _et al._ (2011) K. Mallick, M. Moshe, and H. Orland, J. Phys. A: Math. Theor. 44, 095002 (2011).
  * Wio  _et al._ (2017) H. S. Wio, M. A. Rodríguez, R. Gallego, J. A. Revelli, A. Alés, and R. R. Deza, Front. Phys. 4, 52 (2017).
  * Rodríguez and Wio (2019) M. A. Rodríguez and H. S. Wio, Phys. Rev. E 100, 032111 (2019).
  * Wio  _et al._ (2020a) H. S. Wio, M. A. Rodríguez, and R. Gallego, Chaos: An Interdisciplinary Journal of Nonlinear Science 30, 073107 (2020a).
  * Wio  _et al._ (2020b) H. S. Wio, R. R. Deza, and J. A. Revelli, J. Stat. Mech.: Theory Exp. 2020, 024009 (2020b).
  * Edwards and Wilkinson (1982) S. F. Edwards and D. R. Wilkinson, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 381, 17 (1982).
  * Limmer  _et al._ (2021) D. T. Limmer, C. Y. Gao, and A. R. Poggioli, Eur. Phys. J. B 94, 145 (2021).
  * Jarzynski and Wójcik (2004) C. Jarzynski and D. K. Wójcik, Phys. Rev. Lett. 92, 230602 (2004).
  * Bettencourt (2001) L. M. A. Bettencourt, Phys. Rev. D 63, 045020 (2001).
  * Kerson Huang (1987) H. Kerson Huang,  _Statistical Mechanics_ (John Wiley & Sons, 1987).
  * Parisi and Machta (1989) G. Parisi and J. Machta, Am. J. Phys. 57, 286 (1989).
  * Touchette (2009) H. Touchette, Phys. Rep. 478, 1 (2009).
  * Gambassi and Silva (2012) A. Gambassi and A. Silva, Phys. Rev. Lett. 109, 250602 (2012).


# Degenerations and order of graphs realized by finite abelian groups

Rameez Raja 111Department of Mathematics, National Institute of Technology
Srinagar-190006, Jammu and Kashmir, India. Email<EMAIL_ADDRESS>

Abstract. Let $G_{1}$ and $G_{2}$ be two groups. If a group homomorphism
$\varphi:G_{1}\rightarrow G_{2}$ maps $a\in G_{1}$ into $b\in G_{2}$ such that
$\varphi(a)=b$, then we say $a$ degenerates to $b$ and if every element of
$G_{1}$ degenerates to elements in $G_{2}$, then we say $G_{1}$ degenerates to
$G_{2}$. We discuss degeneration in graphs and show that degeneration in
groups is a particular case of degeneration in graphs. We exhibit some
interesting properties of degeneration in graphs. We use this concept to
present a pictorial representation of graphs realized by finite abelian
groups. We discus some partial orders on the set $\mathcal{T}_{p_{1}\cdots
p_{n}}$ of all graphs realized by finite abelian $p_{r}$-groups, where each
$p_{r}$, $1\leq r\leq n$, is a prime number. We show that each finite abelian
$p_{r}$-group of rank $n$ can be identified with saturated chains of Young
diagrams in the poset $\mathcal{T}_{p_{1}\cdots p_{n}}$. We present a
combinatorial formula which represents the degree of a projective
representation of a symmetric group. This formula determines the number of
different saturated chains in $\mathcal{T}_{p_{1}\cdots p_{n}}$ and the number
of finite abelian groups of different orders.

Keywords: Degenerations, Finite abelian groups, Threshold graph, Partial
order.

AMS subject classification: Primary: 13C70, 05C25.

##  1 Introduction

A notion of degeneration in groups was introduced in [8] to parametrize the
orbits in a finite abelian group under its full automorphism group by a finite
distributive lattice. The authors in [8] were motivated by attempts to
understand the decomposition of the weil representation associated to a finite
abelian group $G$. Note that the sum of squares of the multiplicities in the
Weil representation is the number of orbits in $G\times\hat{G}$ under
automorphisms of a symplectic bicharacter, where $\hat{G}$ denotes the
Pontryagin dual of $G$.

The above combinatorial description is one of the explorations between groups
and combinatorial structures (posets and lattices). There is an intimate
relationship between between groups and other combinatorial structures
(graphs). For example, any graph $\Gamma$ give rise to its automorphism group
whereas any group with its generating set give rise to a realization of a
group as a graph (Cayley graph).

Recently, authors in [13] studied the group-annihilator graph $\Gamma(G)$
realized by a finite abelian group $G$ (viewed as a $\mathbb{Z}$-module) of
different ranks. The vertices of $\Gamma(G)$ are all elements of $G$ and two
vertices $x,y\in G$ are adjacent in $\Gamma(G)$ if and only if
$[x:G][y:G]G=\\{0\\}$, where
$[x:G]=\\{r\in\mathbb{Z}:rG\subseteq\mathbb{Z}x\\}$ is an ideal of a ring
$\mathbb{Z}$. They investigated the concept of creation sequences in
$\Gamma(G)$ and determined the multiplicities of eigenvalues $0$ and $-1$ of
$\Gamma(G)$. Interestingly, they considered orbits of the symmetric group
action: $Aut(\Gamma(G))\times G\longrightarrow G$ and proved that the
representatives of orbits are the Laplacian eigenvalues of $\Gamma(G)$.

There are number of realizations of groups as graphs. The generating graph
[11] realized by a simple group was introduced to get an insight that might
ultimately guide us to a new proof of the classification of simple groups. The
graphs such as power graph [6], intersection graph [4] and the commuting graph
[5] were introduced to study the information contained in the graph about the
group.

Moreover, the realizations of rings as graphs were introduced in [1, 3]. The
aim of considering these realizations of rings as graphs is to study the
interplay between combinatorial and ring theoretic properties of a ring $R$.
This concept was further studied in [16, 18, 19, 20] and was extended to
modules over commutative rings in [21].

The main objective of this work is to investigate some deeper interconnections
between partitions of a number, young diagrams, finite abelain groups, group
homomorphisms, graph homomorphisms, posets and lattices. This investigation
will lead us to develop a theory which is going to simplify the concept of
degeneration of elements in groups and also provide a lattice of finite
abelian groups in which each saturated chain of length $n$ can be identified
with a finite abelian $p_{r}$-group of rank $n$.

This research article is organized as follows. In section 2, we discuss some
results related to degeneration in groups and group-annihilator graphs
realized by finite abelian groups. Section 3 is dedicated to the study of
degenerations in graphs realized by finite abelian groups. We present a
pictorial sketch which illustrates degeneration in graphs. Finally in section
4, we investigate multiple relations on the set $\mathcal{T}_{p_{1}\cdots
p_{n}}$ and furnish the information contained in a locally finite distributive
lattice about finite abelian groups. We provide a combinatorial formula which
represents degree of a projective representation of a symmetric group and the
number of saturated chains from empty set to some non-trivial member of
$\mathcal{T}_{p_{1}\cdots p_{n}}$.

##  2 Preliminaries

Let $\lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{r})$ be a partition of
$n$ denoted by $\lambda\vdash n$, where $n\in\mathbb{Z}_{>0}$ is a positive
integer. For any $\mu\vdash n$, we have an abelian group of order $p^{n}$ and
conversely every abelian group corresponds to some partition of $n$. In fact,
if $H_{\mu,p}=\mathbb{Z}/p^{{\mu}_{1}}\mathbb{Z}\leavevmode\nobreak\
\oplus\leavevmode\nobreak\
\mathbb{Z}/p^{{\mu}_{2}}\mathbb{Z}\leavevmode\nobreak\
\oplus\leavevmode\nobreak\ \cdots\leavevmode\nobreak\
\oplus\leavevmode\nobreak\ \mathbb{Z}/p^{{\mu}_{r}}\mathbb{Z}$ is a subgroup
of $G_{\lambda,p}$
($G_{\lambda,p}=\mathbb{Z}/p^{\lambda_{1}}\mathbb{Z}\oplus\mathbb{Z}/p^{\lambda_{2}}\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/p^{\lambda_{r}}\mathbb{Z}$
is a finite abelian $p$-group), then
$\mu_{1}\leq\lambda_{1},\mu_{2}\leq\lambda_{2},\cdots,\mu_{r}\leq\lambda_{r}$.
If these inequalities holds we write $\mu\subset\lambda$, that is a
“containment order”on partitions. For example, a $p$-group
$\mathbb{Z}/p^{7}\mathbb{Z}\leavevmode\nobreak\ \oplus\leavevmode\nobreak\
\mathbb{Z}/p\mathbb{Z}\leavevmode\nobreak\ \oplus\leavevmode\nobreak\
\mathbb{Z}/p\mathbb{Z}$ is of type $\lambda=(7,1,1)$. The possible types for
its subgroup are: $(7,1,1),(6,1,1),(5,1,1),(4,1,1)$,
$(3,1,1),(2,1,1),(1,1,1),2(7,1),2(6,1),2(5,1),2(4,1),2(3,1),2(2,1),2(1,1),(7),(6),(5),(4)$,
$\noindent(3),(2),2(1)$.

Note that the types $(7,1),(6,1),(5,1),(4,1),(3,1),(2,1),(1,1)$ are appearing
twice in the sequence of partitions for a subgroup.

The authors in [8] have considered the group action: $Aut(G)\times
G\rightarrow G$, where $Aut(G)$ is an automorphism group of $G$ and studied
$Aut(G)\setminus G$, the set of all disjoint $Aut(G)$-orbits in $G$. The group
$\mathbb{Z}/p^{k}\mathbb{Z}$ has $k$ orbits of non-zero elements under the
action of its automorphism group, represented by elements
$1,p,\cdots,p^{k-1}$. We denote orbits of the group action:
$Aut(\mathbb{Z}/p^{k}\mathbb{Z})\times\mathbb{Z}/p^{k}\mathbb{Z}\longrightarrow\mathbb{Z}/p^{k}\mathbb{Z}$
by $\mathcal{O}_{k,p^{m}}$, where $0\leq m\leq k-1$.

Miller [17], Schwachhöfer and Stroppel [22] provided some well known formulae
for the cardinality of the set $Aut(G_{\lambda,p})$ $\setminus$
$G_{\lambda,p}$ .

######  Definition 1.

(Degeneration in groups) [8]. Let $G_{1}$ and $G_{2}$ be two groups, then
$a\in G_{1}$ degenerates to $b\in G_{2}$, if a homomorphism
$\varphi:G_{1}\longrightarrow G_{2}$ maps $a$ into $b$ such that
$\varphi(a)=b$.

The following result provide a characterization for degenerations of elements
of the group $\mathbb{Z}/p^{k}\mathbb{Z}$ to elements of the group
$\mathbb{Z}/p^{l}\mathbb{Z}$, where $k\leq l$.

######  Lemma 2.

[8]. $p^{r}u\in\mathcal{O}_{k,p^{r}}$ in $\mathbb{Z}/p^{k}\mathbb{Z}$
degenerates to $p^{s}v\in\mathcal{O}_{l,p^{s}}$ in
$\mathbb{Z}/p^{l}\mathbb{Z}$ if and only if $r\leq s$ and $k-r\geq l-s$, where
$u,v$ are relatively prime to $p$, $r<k$ and $s<l$. If in addition
$p^{s}v\in\mathcal{O}_{l,p^{s}}$ degenerates to
$p^{r}u\in\mathcal{O}_{k,p^{r}}$, then $k=l$ and $r=s$.

By Lemma 2, it is easy to verify that degeneracy is a partial order relation
on the set of all orbits of non-zero elements in $\mathbb{Z}/p^{k}\mathbb{Z}$.
The diagrammatic representation (Hasse diagram) of the set
$Aut(\mathbb{Z}/p^{k}\mathbb{Z})\setminus\mathbb{Z}/p^{k}\mathbb{Z}$ with
respect to degeneracy, which is called a fundamental poset is presented in
[Figure 1 [8]].

Let $a=(a_{1},a_{2},\cdots,a_{r})\in G_{\lambda,p}$, the ideal of a in
$Aut(G_{\lambda,p})$ $\setminus$ $G_{\lambda,p}$ denoted by $I(a)$ is the
ideal generated by orbits of non-zero coordinates
$a_{i}\in\mathbb{Z}/p^{\lambda_{i}}\mathbb{Z}$. One of the explorations
between ideals of posets, partitions and orbits of finite abelian groups is
the following interesting result.

######  Theorem 3.

[8]. Let $\lambda$ and $\mu$ be any two given partitions and $a\in
G_{\lambda,p}$, $b\in G_{\mu,p}$. Then $a$ degenerates to $b$ if and only if
$I(b)\subset I(a)$.

The enumeration of orbits as ideals, first as counting ideals in terms of
their boundaries, and the second as counting them in terms of anti chains of
maximal elements is presented in [Example 6.1, 6.2 [8]].

Please see sections 7 and 8 of [8] for results related to embedding of the
lattice of orbits of $G_{\lambda,p}$ into the lattice of characteristic
subgroups of $G_{\lambda,p}$, formula for the order of the characteristic
subgroup associated to an orbit, computation of a monic polynomial in $p$
(with integer coefficients) using mobius inversion formula representing
cardinality of the orbit in $G_{\lambda,p}$.

Let $\Gamma=(V,E)$ be a simple connected graph and let $\Gamma_{1}$ and
$\Gamma_{2}$ be two simple connected graphs, recall a mapping
$\phi:V(\Gamma_{1})\rightarrow V(\Gamma_{2})$ is a homomorphism if it
preserves edges, that is, for any edge $(u,v)$ of $\Gamma_{1}$,
$(\phi(u),\phi(v))$ is an edge of $\Gamma_{2}$, where $u,v\in V(\Gamma_{1})$.
A homomorphism $\phi:V(\Gamma_{1})\rightarrow V(\Gamma_{2})$ is faithful when
there is an edge between two pre images $\phi^{-1}(u)$ and $\phi^{-1}(u)$ such
that $(u,v)$ is an edge of $\Gamma_{2}$, a faithful bijective homomorphism is
an isomorphism and in this case we write $\Gamma_{1}\cong\Gamma_{2}$. An
isomorphism from $\Gamma$ to itself is an automorphism of $\Gamma$, it is well
known that set of automorphisms of $\Gamma$ forms a group under composition,
we denote the group of automorphisms of $\Gamma$ by $Aut(\Gamma)$.
Understanding the automorphism group of a graph is a guiding principle for
understanding objects by their symmetries.

Consider the group action: $Aut(\Gamma)\leavevmode\nobreak\
acting\leavevmode\nobreak\ on\leavevmode\nobreak\ V(\Gamma)$ by some
permutation of $Aut(\Gamma)$, that is,

$Aut(\Gamma)\times V(\Gamma)\rightarrow V(\Gamma)$,

$\sigma(v)=u$,

where $\sigma\in Aut(\Gamma)$ and $v,u\in V(\Gamma)$ are any two vertices of
$\Gamma$. This group action is called a symmetric action [13].

Consider a finite abelian non-trivial group $G$ with identity element $0$ and
view $G$ as a $\mathbb{Z}$-module. For $a\in G$, set
$[a:G]=\\{x\in\mathbb{Z}\leavevmode\nobreak\ |\leavevmode\nobreak\
xG\subseteq\mathbb{Z}a\\}$, which clearly is an ideal of $\mathbb{Z}.$ For
$a\in G$, $G/\mathbb{Z}a$ is a $\mathbb{Z}$-module. So, $[a:G]$ is a
annihilator of $G/\mathbb{Z}a$, $[a:G]$ is called a $a$-annihilator of $G.$
Also, an element $a$ is called an ideal-annihilator of $G$ if there exists a
non-zero element $b$ of $G$ such that $[a:G][b:G]G=\\{0\\}$, where
$[a:G][b:G]$ denotes the product of ideals of $\mathbb{Z}$. The element $0$ is
a trivial ideal-annihilator of $G$, since $[0:G][b:G]G=ann(G)[b:G]G=\\{0\\}$,
$ann(G)$ is an annihilator of $G$ in $\mathbb{Z}$.

Given an abelian group $G$, the group-annihilator graph is defined to be the
graph $\Gamma(G)=(V(\Gamma(G))$, $E(\Gamma(G)))$ with vertex set
$V(\Gamma(G))=G$ and for two distinct $a,b\in V(\Gamma(G))$, the vertices $a$
and $b$ are adjacent in $\Gamma(G)$ if and only if $[a:G][b:G]G=\\{0\\}$, that
is, $E(G)=\\{(a,b)\in G\times G:[a:G][b:G]G=\\{0\\}\\}$.

For a cyclic group $G=\mathbb{Z}/p^{n}\mathbb{Z}$ ($n\geq 1$), it is easy to
verify that the orbits of the action: $Aut(G)\times G\longrightarrow G$ are
same as the orbits of the symmetric action: $Aut(\Gamma(G))\times
G\longrightarrow G$ which are given as follows,

$\mathcal{O}_{n,p^{i}}=\\{p^{i}\alpha(mod\leavevmode\nobreak\
p^{n})\mid\alpha\in\mathbb{Z},(\alpha,p)=1\\}$,

where $i\in[0,n]$. Furthermore, for $0\leq i<j\leq n$, $p^{i}\alpha\equiv
p^{j}\alpha^{\prime}(mod\leavevmode\nobreak\ p^{n})\text{ where
}(\alpha,p)=1\text{ and }(\alpha^{\prime},p)=1$. Consequently, we have for
$i\neq j$, $\mathcal{O}_{n,p^{i}}\cap\mathcal{O}_{n,p^{j}}=\emptyset$.

Any element $a\in\mathbb{Z}/p^{n}\mathbb{Z}$ can be expressed as,

$a\equiv
p^{n-1}b_{1}+p^{n-2}b_{2}+\cdots+pb_{n-1}+b_{n}(mod\leavevmode\nobreak\
p^{n})$,

where $b_{i}\in[1,p-1]$. If $a\in\mathcal{O}_{n,1}$, then $b_{n}\neq 0.$ So,
$|\mathcal{O}_{n,1}|=p^{n-1}(p-1)=\phi(p^{n})$. If
$a^{\prime}\in\mathcal{O}_{n,p}$, then for some $a\in\mathcal{O}_{n,1}$
$a^{\prime}=pa$, that is, $b_{n}\neq 0$, so
$|\mathcal{O}_{n,p}|=\frac{\phi(p^{n})}{p}$. Similarly, for $i\in[0,n]$, we
have $|\mathcal{O}_{n,p^{i}}|=\frac{\phi(p^{n})}{p^{i}}$.

######  Proposition 2.1.

[13]. Let $G=\mathbb{Z}/p^{n}\mathbb{Z}$ be a cyclic group of order $p^{n}$,
where $n\geq 2$. Then for each $a\in\mathcal{O}_{n,p^{i}}$ with $i\in[1,n]$,
the $a-$annihilator of $G$ is $[a:G]=p^{i}\mathbb{Z}$

Thus if we consider the symmetric group action: $Aut(\Gamma(G))\times
G\longrightarrow G$, then for $G=\mathbb{Z}/p^{n}\mathbb{Z}$, the group-
annihilator graph realized by $G$ is defined as
$\Gamma(G)=(V(\Gamma(G)),E(\Gamma(G)))$, where
$V(\Gamma(G))=\mathbb{Z}/p^{n}\mathbb{Z}$ and two vertices
$u\in\mathcal{O}_{n,p^{i}}$, $v\in\mathcal{O}_{n,p^{j}}$ are adjacent in
$\Gamma(G)$ if and only if $i+j\geq n$.

Therefore, from the above observation it follows that the vertices of the
graph $\Gamma(G)$ are parametrized by representatives of orbits of the group
action: $Aut(\Gamma(G))\times G\longrightarrow G$. Thus an element
$0\in\mathcal{O}_{n,p^{n}}$ of $G$ is adjacent to all vertices in $\Gamma(G)$,
elements $a\in\mathcal{O}_{n,1}$ which are prime to order of $G$ are adjacent
to $0$ only in $\Gamma(G)$. Furthermore, elements of the orbit
$\mathcal{O}_{n,p}$ are adjacent to $0$ and elements of the orbit
$\mathcal{O}_{n,p^{n-1}}$, elements of the orbit $\mathcal{O}_{n,p^{2}}$ are
adjacent to $0$ and elements of the orbits $\mathcal{O}_{n,p^{n-1}}$,
$\mathcal{O}_{n,p^{n-2}}$. Thus, for $k\geq 1$, elements of the orbit
$\mathcal{O}_{n,p^{k}}$ are adjacent to elements of the orbits
$\mathcal{O}_{n,p^{n-k}}$,
$\mathcal{O}_{n,p^{n-k+1}},\cdots,\mathcal{O}_{n,p^{n-1}}$,
$\mathcal{O}_{n,p^{n}}$.

######  Theorem 4.

[13]. Let $n$ be a positive integer. Then for the $p$-group
$G=(\mathbb{Z}/p^{n}\mathbb{Z})^{\ell}$ of rank $\ell\geq 2,$ and
$(a_{1},\ldots,a_{l})\in G$, the $(a_{1},\ldots,a_{l})$-annihilator of $G$ is
$p^{n}\mathbb{Z}.$ In particular the corresponding group-annihilator graph
realized by $G$ is a complete graph.

Note that the action of $Aut(\Gamma((\mathbb{Z}/p\mathbb{Z})^{\ell}))$ on
$(\mathbb{Z}/p\mathbb{Z})^{\ell}$ is transitive, since an automorphism of
$\Gamma((\mathbb{Z}/p\mathbb{Z})^{\ell})$ map any vertex to any other vertex
and this does not place any restriction on where any of the other $p^{\ell}-1$
vertices are mapped, as they are all mutually connected in
$\Gamma((\mathbb{Z}/p\mathbb{Z})^{\ell})$. This implies
$Aut(\Gamma((\mathbb{Z}/p\mathbb{Z})^{\ell}))\setminus(\mathbb{Z}/p\mathbb{Z})^{\ell}$
is a single orbit of order $p^{\ell}$.

For more information regarding $a-$annihilators, $(a,b)-$annihilators and
$(a_{1},a_{2},\cdots,a_{l})$ $-$annihilators of finite abelian $p$-groups,
please see section 3 of [13].

We conclude this section by an example which illustrates the parametrization
of vertices of the group-annihilator graph $\Gamma(G)$ by representatives of
orbits of the symmetric action on $G$.

######  Example 5.

Let $G=\mathbb{Z}/2^{4}\mathbb{Z}$ be a finite abelian. Consider the group
action: $Aut(\Gamma(G))\times G\longrightarrow G$. The orbits of this action
are: $\mathcal{O}_{4,2^{4}}=\\{0\\}$, $\mathcal{O}_{4,1}=\\{1,3,5,7\\}$,
$\mathcal{O}_{4,2}=\\{2,6,10,14\\}=\\{2a\leavevmode\nobreak\
|\leavevmode\nobreak\ (a,2)=1\\}$,
$\mathcal{O}_{4,2^{2}}=\\{4,12\\}=\\{2^{2}a\leavevmode\nobreak\
|\leavevmode\nobreak\ (a,2)=1\\}$ and
$\mathcal{O}_{4,2^{3}}=\\{8\\}=\\{2^{3}a\leavevmode\nobreak\
|\leavevmode\nobreak\ (a,2)=1\\}$. Note that orbits of elements $3,5,7$ are
same as the orbit of $1$, orbits of $6,10,14$ are same as the orbit of $2$ and
orbit of $12$ is same as the orbit of $4$. Therefore, the group $G$ has $4$
orbits of nonzero elements under the action of $Aut(\Gamma(G))$ represented by
$1,2,2^{2},2^{3}$. The group-annihilator graph realized by $G$ with its orbits
is shown in Figure (1).

Figure 1: $\Gamma(\mathbb{Z}/2^{4}\mathbb{Z})$ with its orbits

##  3 Degeneration in graphs

This section is devoted to the study of degeneration in graphs. We show that
every group homomorphism is a graph homomorphism. We employ the methods of
degeneration in graphs to simply the techniques used to establish
degenerations of elements in finite abelian groups [8].

As far as groups are concerned, there are always homomorphisms (trivial
homomorphisms) from one group to another. Any source group (a group where from
we have the map) can be mapped by a homomorphism into target group (a group
where the elements are mapped) by simply sending all of its elements to the
identity of the target group. In fact, the study of kernels is very important
in algebraic structures. In the context of simple graphs, the notion of a
homomorphism is far more restrictive. Indeed, there need not be a homomorphism
between two graphs, and these cases are as much a part of the theory as those
where homomorphisms do exist. There are other categories where homomorphisms
do not always exist between two objects, for example, the category of bounded
lattices or that of semi-groups.

The answer to the question that “every group homomorphism is a graph
homomorphism” is affirmative, and the same is discussed in the following
result. Note that the orbits of elements of actions (automorphism group and
symmetric) on finite abelian $p$-group of rank one coincide and it can be
explored further on abelian $p$-groups of different ranks.

######  Proposition 3.1.

Every group homomorphism which maps elements from orbits
$\mathcal{O}_{k,p^{i}}$ to orbits $\mathcal{O}_{l,p^{j}}$ is a graph
homomorphism, where $1\leq i\leq k$, $1\leq j\leq l$ and $k\leq l$.

###### Proof.

The group homomorphisms are uniquely determined by the image of unity element
in the target group and order of the element divides order of unity in the
source group. Let $\tau(a)$ be the image of unity in the target group.
Therefore, we have $\tau(a)=a_{1},a_{2},\cdots,a_{p^{k}}$, where
$a_{1},a_{p},\cdots,a_{p^{k}}$ are elements of orbits, $\mathcal{O}_{l,1}$,
$\mathcal{O}_{l,p}$, $\cdots$, $\mathcal{O}_{l,p^{k}}$. Note that $k\leq l$,
therefore we have the following inequalities concerning the cardinalities of
obits,

$|\mathcal{O}_{k,1}|\leq|\mathcal{O}_{l,1}|$,

$|\mathcal{O}_{k,p}|\leq|\mathcal{O}_{l,p}|$,

⋮

$|\mathcal{O}_{k,p^{k}}|\leq|\mathcal{O}_{l,p^{k}}|$.

If $\tau(a)\in\mathcal{O}_{l,1}$, then under the monomorphism the elements of
orbits are mapped as,

$\mathcal{O}_{k,1}\xhookrightarrow{\leavevmode\nobreak\
1-1\leavevmode\nobreak\ }\mathcal{O}_{l,1}$,

$\mathcal{O}_{k,p}\xhookrightarrow{\leavevmode\nobreak\
1-1\leavevmode\nobreak\ }\mathcal{O}_{l,p}$,

⋮

$\mathcal{O}_{k,p^{k-1}}\xhookrightarrow{\leavevmode\nobreak\
1-1\leavevmode\nobreak\ }\mathcal{O}_{l,p^{k-1}}$,

$\mathcal{O}_{k,p^{k}}\xhookrightarrow{\leavevmode\nobreak\
1-1\leavevmode\nobreak\ }\mathcal{O}_{l,p^{l}}$.

If $\tau(a)\in\mathcal{O}_{l,p}$, then elements of orbits are mapped as,

$\mathcal{O}_{k,1}\twoheadrightarrow\mathcal{O}_{l,p}$,

$\mathcal{O}_{k,p}\twoheadrightarrow\mathcal{O}_{l,p^{2}}$,

⋮

$\mathcal{O}_{k,p^{k-1}}\twoheadrightarrow\mathcal{O}_{l,p^{k}}$,

$\mathcal{O}_{k,p^{k}}\twoheadrightarrow\mathcal{O}_{l,p^{l}}$.

Thus it follows that if $\tau(a)\in\mathcal{O}_{l,p^{t}}$ for $(0\leq t\leq
k-1)$, then every element of the orbit $\mathcal{O}_{k,p^{t}}$ is mapped to
elements of the orbit $\mathcal{O}_{l,p^{t+1}}$.

Under the symmetric action the orbits of vertices are same as the orbits
listed above. Note that the vertices of the orbit $\mathcal{O}_{k,1}$ are only
adjacent to the vertex in $\mathcal{O}_{k,p^{k}}$, vertices of the orbit
$\mathcal{O}_{k,p}$ are adjacent to vertices in $\mathcal{O}_{k,p^{k}}$ and
$\mathcal{O}_{k,p^{k-1}}$ and so on. Thus if $\tau(a)\in\mathcal{O}_{l,1}$,
then for $0\leq i\leq j\leq k$, every edge
$(u,v)\in\mathcal{O}_{k,p^{i}}\times\mathcal{O}_{k,p^{j}}$ is mapped to edges
$(\tau(u),\tau(v))\in\mathcal{O}_{l,p^{r}}\times\mathcal{O}_{l,p^{s}}$, where
$0\leq r\leq s\leq l$. Therefore $\tau$ is a graph homomorphism. Similarly it
can be verified that all other group homomorphisms are graph homomorphisms,
since the adjacencies are preserved under all group homomorphisms. ∎

######  Remark 3.2.

The converse of the preceding result is not true, that is, a graph
homomorphism between two graphs realised by some groups need not to be a group
homomorphism. To illustrate this we consider the “distribution of edges in
orbits”. Theoretically, distribution of edges is carried out in a way that for
sufficiently large $l$, a graph homomorphism is acting on vertices in orbits
$\mathcal{O}_{k,p^{k}}$, $\mathcal{O}_{k,1}$ such that
$\mathcal{O}_{k,p^{k}}\xhookrightarrow{identity}\mathcal{O}_{l,p^{l}}$,
$\mathcal{O}_{k,p^{k-1}}\xhookrightarrow{identity}\mathcal{O}_{l,p^{k-1}}$,
$\cdots$, $\mathcal{O}_{k,p}\xhookrightarrow{identity}\mathcal{O}_{l,p}$. Some
vertices of $\mathcal{O}_{k,1}$ are mapped to itself in $\mathcal{O}_{l,1}$
whereas the remaining are mapped to vertices in $\mathcal{O}_{l,p}$. So, under
the above distribution some edges in
$\mathcal{O}_{k,p^{k}}\times\mathcal{O}_{k,1}$ are mapped to edges in
$\mathcal{O}_{l,p^{l}}\times\mathcal{O}_{l,1}$, whereas the remaining edges in
$\mathcal{O}_{k,p^{k}}\times\mathcal{O}_{k,1}$ are mapped to edges in
$\mathcal{O}_{l,p^{l}}\times\mathcal{O}_{l,p}$. Thus if $x\neq y$ are two
elements of $\mathcal{O}_{k,1}$ such that $x$ is mapped to
$x^{\prime}\in\mathcal{O}_{l,1}$ and $y$ is mapped to
$y^{\prime}\in\mathcal{O}_{l,p}$, then the following equation may have no
solution,

$x+y(mod\leavevmode\nobreak\
p^{k})=x^{\prime}+y^{\prime}(mod\leavevmode\nobreak\ p^{l})$.

######  Definition 6.

Let $\Gamma_{1}$ and $\Gamma_{2}$ be two simple graphs. Then $(a,b)\in
E(\Gamma_{1})$ degenerates to $(u,v)\in E(\Gamma_{2})$ if there exists a
homomorphism $\varphi:V(\Gamma_{1})\longrightarrow V(\Gamma_{2})$ such that
$\varphi(a,b)=(u,v)$. If every edge of $\Gamma_{1}$ degenerates to edges in
$\Gamma_{2}$, then we say that $\Gamma_{1}$ degenerates to $\Gamma_{2}$.

Recall that an independent part (independent set) in a graph $\Gamma$ is a set
of vertices of $\Gamma$ such that for every two vertices, there is no edge in
$\Gamma$ connecting the two. Also, the complete part (complete subgraph) in a
graph $\Gamma$ is a set of vertices in $\Gamma$ such that there is an edge
between every pair of vertices in $\Gamma$.

The simplified form of Lemma (2) is presented in the following result. We
adapted the definition of degeneration in groups and make it to work for
graphs which are realized by finite abelian groups.

######  Theorem 7.

If under any graph homomorphism $\mathcal{O}_{k,p^{k}}$ is the only vertex
mapped to $\mathcal{O}_{l,p^{l}}$, then the pair
$(p^{r}u,p^{s}u)\in\mathcal{O}_{k,p^{r}}\times\mathcal{O}_{k,p^{s}}$
degenerates to
$(p^{r^{\prime}}u,p^{s^{\prime}}u)\in\mathcal{O}_{l,p^{r^{\prime}}}\times\mathcal{O}_{l,p^{s^{\prime}}}$
if and only if $r\leq r^{\prime}$ and $s\leq s^{\prime}$, where $u$ is
relatively prime to $p$ and $k\leq l$.

###### Proof.

In setting of the symmetric group action on finite abelain $p$-groups of rank
one, let $\mathcal{O}_{k,p^{r}}$, $\mathcal{O}_{k,p^{s}}$ be orbits
represented by elements $p^{r}$ and $p^{s}$ of the source group and
$\mathcal{O}_{l,p^{r^{\prime}}}$, $\mathcal{O}_{l,p^{s^{\prime}}}$ be orbits
represented by elements $p^{r^{\prime}}$ and $p^{s^{\prime}}$ of the target
group, where $0\leq r,\leavevmode\nobreak\ s\leq k-1$ and $0\leq
r^{\prime},\leavevmode\nobreak\ s^{\prime}\leq l-1$. We consider the cases
hereunder.

Case I: $k=l=2t$, $t\in\mathbb{Z}_{>0}$. Then the independent and complete
parts of the graph realised by a source group is
$X=\dot{\bigcup}_{i=0}^{t-1}\mathcal{O}_{k,p^{i}}$ and
$Y=\dot{\bigcup}_{i=0}^{t-1}\mathcal{O}_{k,p^{t+j}}$, where each element of
both $X$ and $Y$ are connected to $\mathcal{O}_{k,p^{k}}=\\{0\\}$. Similarly,
$X^{\prime}=\dot{\bigcup}_{i=0}^{t-1}\mathcal{O}_{l,p^{i}}$ and
$Y^{\prime}=\dot{\bigcup}_{j=0}^{t-1}\mathcal{O}_{l,p^{t+j}}$ represents the
independent and complete parts of the graph realized by a target group, where
each element of both $X^{\prime}$ and $Y^{\prime}$ are connected to
$\mathcal{O}_{l,p^{l}}=\\{0\\}$.

Let $x\in X$. If $x\in\mathcal{O}_{k,1}$, then as discussed above, $x$ is
adjacent to $\mathcal{O}_{k,p^{k}}$ only. On the other hand, if
$x\in\mathcal{O}_{k,p^{i}}$ for $1\leq i\leq t-1$, then $x$ is adjacent to all
elements of the set $\dot{\bigcup}_{n=i}^{0}\mathcal{O}_{k,p^{k-n}}\subset Y$.
Moreover, if $x^{\prime}\in\mathcal{O}_{l,1}$, then $x^{\prime}$ is adjacent
to $\mathcal{O}_{l,p^{l}}$ whereas if $x^{\prime}\in\mathcal{O}_{l,p^{j}}$ for
$1\leq j\leq t-1$, then $x^{\prime}$ is adjacent to all elements of the set
$\dot{\bigcup}_{m=j}^{0}\mathcal{O}_{l,p^{l-m}}\subset Y^{\prime}$. Under any
given graph homomorphism $\tau$, the images of relations in
$X\times\mathcal{O}_{k,p^{k}}$, $X\times Y$ and $Y\times\mathcal{O}_{k,p^{k}}$
are in $X^{\prime}\times\mathcal{O}_{l,p^{l}}$, $X^{\prime}\times Y^{\prime}$
and $Y^{\prime}\times\mathcal{O}_{l,p^{l}}$. Let $(a,b)\in
X\times\mathcal{O}_{k,p^{k}}\bigcup X\times Y\bigcup
Y\times\mathcal{O}_{k,p^{k}}$. Suppose $(a,b)$ degenerates to some
$(a^{\prime},b^{\prime})\in X^{\prime}\times\mathcal{O}_{l,p^{l}}\bigcup
X^{\prime}\times Y^{\prime}\bigcup Y^{\prime}\times\mathcal{O}_{l,p^{l}}$ . If
$\tau$ is group homomorphism such that $\tau(1)\in\mathcal{O}_{l,1}$, then

$\mathcal{O}_{k,1}\times\mathcal{O}_{k,p^{k}}\xhookrightarrow{1-1}\mathcal{O}_{l,1}\times\mathcal{O}_{l,p^{l}}$,

$\mathcal{O}_{k,p}\times\mathcal{O}_{k,p^{k}}\xhookrightarrow{1-1}\mathcal{O}_{l,p}\times\mathcal{O}_{l,p^{l}}$,

$\mathcal{O}_{k,p}\times\mathcal{O}_{k,p^{k-1}}\xhookrightarrow{1-1}\mathcal{O}_{l,p}\times\mathcal{O}_{l,p^{l-1}}$,

⋮

If $\tau(1)\in\mathcal{O}_{l,p}$, then

$\mathcal{O}_{k,1}\times\mathcal{O}_{k,p^{k}}\twoheadrightarrow\mathcal{O}_{l,p}\times\mathcal{O}_{l,p^{l}}$,

$\mathcal{O}_{k,p}\times\mathcal{O}_{k,p^{k}}\twoheadrightarrow\mathcal{O}_{l,p^{2}}\times\mathcal{O}_{l,p^{l}}$,

$\mathcal{O}_{k,p}\times\mathcal{O}_{k,p^{k-1}}\twoheadrightarrow\mathcal{O}_{l,p^{2}}\times\mathcal{O}_{l,p^{l-1}}$,

⋮

If $\tau(1)$ lies in any other orbit of $X^{\prime}\bigcup Y^{\prime}$, then
as above we have the mapping of edges to edges. Thus for any group
homomorphism which maps
$(p^{r}u,p^{s}u)\in\mathcal{O}_{k,p^{r}}\times\mathcal{O}_{k,p^{s}}$ to
$(p^{r^{\prime}}u,p^{s^{\prime}}u)\in\mathcal{O}_{l,p^{r^{\prime}}}\times\mathcal{O}_{l,p^{s^{\prime}}}$,
the relations $r\leq r^{\prime}$ and $s\leq s^{\prime}$ are verified.

Now, suppose $\tau$ is not a group homomorphism but a graph homomorphism.
Assume without loss of generality that under $\tau$,
$A\times\mathcal{O}_{k,p^{k}}\xhookrightarrow{1-1}A^{\prime}\times\mathcal{O}_{l,p^{l}}$,
where $A\subset\mathcal{O}_{k,1}\subset X$ and
$A^{\prime}\subset\mathcal{O}_{l,1}\subset X^{\prime}$ are proper subsets of
$X$ and $X^{\prime}$. Moreover,

$\mathcal{O}_{k,1}\setminus
A\times\mathcal{O}_{k,p^{k}}\bigcup\mathcal{O}_{k,p}\times\mathcal{O}_{k,p^{k}}\bigcup\mathcal{O}_{k,p}\times\mathcal{O}_{k,p^{k-1}}\twoheadrightarrow\mathcal{O}_{l,p}\times\mathcal{O}_{l,p^{l-1}}\bigcup\mathcal{O}_{l,p}\times\mathcal{O}_{l,p^{l}}$,

$\mathcal{O}_{k,p^{2}}\times\mathcal{O}_{k,p^{k}}\xhookrightarrow{1-1}\mathcal{O}_{l,p^{2}}\times\mathcal{O}_{l,p^{l}}$,

$\mathcal{O}_{k,p^{2}}\times\mathcal{O}_{k,p^{k-1}}\xhookrightarrow{1-1}\mathcal{O}_{l,p^{2}}\times\mathcal{O}_{l,p^{l-1}}$,

$\mathcal{O}_{k,p^{2}}\times\mathcal{O}_{k,p^{k-2}}\xhookrightarrow{1-1}\mathcal{O}_{l,p^{2}}\times\mathcal{O}_{l,p^{l-2}}$,

⋮

Thus, for $\tau$, we observe that the relations $r\leq r^{\prime}$ and $s\leq
s^{\prime}$ hold. Similarly these relations can be verified for other graph
homomorphisms.

Suppose to the contrary that $r>r^{\prime}$ and $s>s^{\prime}$. Then $(a,b)$
does not degenerates to $(a^{\prime},b^{\prime})$, since by Lemma (2), $a$ and
$b$ degenerates to $a^{\prime}$ and $b^{\prime}$ if and only if $r\leq
r^{\prime}$ and $s\leq s^{\prime}$, therefore, a contradiction. Further, if
under any graph homomorphism the elements of orbits
$\mathcal{O}_{k,p^{r}}\times\mathcal{O}_{k,p^{s}}$ are mapped to elements of
$\mathcal{O}_{l,p^{r^{\prime}}}\times\mathcal{O}_{l,p^{s^{\prime}}}$, then it
follows that for some $1\leq s\leq k-1$, $\mathcal{O}_{k,p^{s}}$ is mapped to
$\mathcal{O}_{l,p^{l}}$, again a contradiction.

Case II: $k=l=2t+1$, $t\in\mathbb{Z}_{>0}$. The independent and complete parts
of the graph realised by source and target groups are
$X=\dot{\bigcup}_{i=0}^{t}\mathcal{O}_{k,p^{i}}$,
$Y=\dot{\bigcup}_{j=1}^{t+1}\mathcal{O}_{l,p^{t+j}}$ and
$X^{\prime}=\dot{\bigcup}_{i=0}^{t}\mathcal{O}_{l,p^{i}}$,
$Y^{\prime}=\dot{\bigcup}_{j=0}^{t+1}\mathcal{O}_{l,p^{t+j}}$. Rest of the
proof for this case follows by the same argument which we discussed above for
the even case.

Finally, if we consider the cases $(k,l)=(2t,2t+1)$ or $(k,l)=(2t+1,2t)$, then
these cases can be handled in the same manner as above. ∎

Figure 2: Pictorial sketch of degeneration

Note that in Figure (2), the graph on the left hand side is the graph realized
by $\mathbb{Z}/2^{3}\mathbb{Z}$ and the graph on the right hand side is
realized by $\mathbb{Z}/2^{5}\mathbb{Z}$.

##  4 Partial orders on $\mathcal{T}_{p_{1}\cdots p_{n}}$

In this section, we study some relations on the set $\mathcal{T}_{p_{1}\cdots
p_{n}}$ of all graphs realized by finite abelian $p_{r}$-groups of rank $1$,
where each $p_{r}$, $1\leq r\leq n$, is a prime number. We discuss equivalent
forms of the partial order “degeneration” on $\mathcal{T}_{p_{1}\cdots p_{n}}$
and obtain a locally finite distributive lattice of finite abelian groups.

Threshold graphs play an essential role in graph theory as well as in several
applied areas which include psychology and computer science [12]. These graphs
were introduced by Chvátal and Hammer [7] and Henderson and Zalcstein [10].

A vertex in a graph $\Gamma$ is called dominating if it is adjacent to every
other vertex of $\Gamma$. A graph $\Gamma$ is called a threshold graph if it
is obtained by the following procedure.

Start with $K_{1}$, a single vertex, and use any of the following steps, in
any order, an arbitrary number of times.

(i) Add an isolated vertex.

(ii) Add a dominating vertex, that is, add a new vertex and make it adjacent
to each existing vertex.

It is always interesting to determine the classes of threshold graphs, since
we may represent a threshold graph on $n$ vertices using a binary code
$(b_{1},b_{2},\cdots,b_{n})$, where $b_{i}=0$ if vertex $v_{i}$ is being added
as an isolated vertex and $b_{i}=1$ if $v_{i}$ is being added as a dominating
vertex. Furthermore, using the concept of creation sequences we establish the
nullity, multiplicity of some non-zero eigenvalues and the Laplacian
eigenvalues of a threshold graph. The Laplacian eigenvalues of $\Gamma$ are
the eigenvalues of a matrix $D(\Gamma)-A(\Gamma)$, where $D(\Gamma)$ is the
diagonal matrix of vertex degrees and $A(\Gamma)$ is the familiar $(0,1)$
adjacency matrix of $\Gamma$.

The authors in [13] confirmed that the graph realised by a finite abelian
$p$-group of rank $1$ is a threshold graph. In fact, they proved the following
intriguing result for a finite abelain $p$-groups of rank $1$.

######  Theorem 8.

[13]. If $G$ is a finite abelian $p$-group of rank $1$, then $\Gamma(G)$ is a
threshold graph.

Let $p_{1}<p_{2}<\cdots<p_{n}$ be a sequence of primes and let
$\lambda_{i}=(\lambda_{i,1},\lambda_{i,2},\cdots,\lambda_{i,n})$ be sequence
of partitions of positive integers, where $1\leq i\leq n$. For each prime
$p_{t}$, where $1\leq t\leq n$, the sequences of finite abelian $p_{t}$-groups
with respect to partitions $\lambda_{i,1},\lambda_{i,2},\cdots,\lambda_{i,n}$
are listed as follows,

$G_{\lambda_{1},p_{1}}=\mathbb{Z}/p_{1}^{\lambda_{1,1}}\mathbb{Z}\oplus\mathbb{Z}/p_{1}^{\lambda_{1,2}}\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/p_{1}^{\lambda_{1,n}}\mathbb{Z}$,

$G_{\lambda_{2},p_{2}}=\mathbb{Z}/p_{2}^{\lambda_{2,1}}\mathbb{Z}\oplus\mathbb{Z}/p_{2}^{\lambda_{2,2}}\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/p_{2}^{\lambda_{2,n}}\mathbb{Z}$,

⋮

Fix a prime $p_{r}$, where $1\leq r\leq n$. Then for each distinct power
$\lambda_{i,j}$, $1\leq i,j\leq n$, it follows from Theorem (8), that members
of the sequence of graphs realised by a sequence of finite abelian
$p_{r}$-groups of rank $1$ are threshold graphs. The sets of orbits of
symmetric group action on sequence of finite abelian $p_{r}$-groups
$\mathbb{Z}/p_{r}^{\lambda_{r,1}}\mathbb{Z},\mathbb{Z}/p_{r}^{\lambda_{r,2}}\mathbb{Z},\cdots,\mathbb{Z}/p_{r}^{\lambda_{r,n}}\mathbb{Z}$
of rank $1$ are:

$\\{\mathcal{O}_{r,1},\mathcal{O}_{r,p_{r}^{1}}\\}$,

$\\{\mathcal{O}_{r,1},\mathcal{O}_{r,p_{r}^{1}},\mathcal{O}_{r,p_{r}^{2}}\\}$,

$\\{\mathcal{O}_{r,1},\mathcal{O}_{r,p_{r}^{1}},\mathcal{O}_{r,p_{r}^{2}},\mathcal{O}_{r,p_{r}^{3}}\\}$,

⋮

Note that, $\lambda_{r,1}=1,\lambda_{r,2}=2,\lambda_{r,3}=3,\cdots$, in the
above sequence of finite abelian$p_{r}$-groups.

Thus for each prime $p_{r}$ and positive integer $\lambda_{i,j}$, we have
sequences of threshold graphs realised by sequences of abelian $p_{r}$-groups.

The degree sequence of a graph $\Gamma$ is given by
$\pi(\Gamma)=(d_{1},d_{2},\cdots,d_{n})$, which is the non-increasing sequence
of non-zero degrees of vertices of $\Gamma$.

For a graph $\Gamma$ of order $n$ and size $m$, let
$d=[d_{1},d_{2},\cdots,d_{n}]$ be a sequence of non-negative integers arranged
in non-increasing order, which we refer to as a partition of $2m$. Define the
transpose of the partition as $d^{*}=[d_{1}^{*},d_{2}^{*},\cdots,d_{r}^{*}]$,
where $d_{j}^{*}=|\\{d_{i}:d_{i}\geq j\\}|$, $j=1,2,\cdots,r$. Therefore
$d_{j}^{*}$ is the number of $d_{i}$’s that are greater than equal to $j$.
Recall from [2] that a sequence $d^{*}$ is called the conjugate sequence of
$d$. The another interpretation of a conjugate sequence is the Ferrer’s
diagram (or Young diagram) denoted by $Y(d)$ corresponding to
$d_{1},d_{2},\cdots,d_{n}$ consists of $n$ left justified rows of boxes, where
the $i^{th}$ row consists of $d_{i}$ boxes (blocks), $i=1,2,\cdots,n$. Note
that $d_{i}^{*}$ is the number of boxes in the $i^{th}$ column of the Young
diagram with $i=1,2,\cdots,r$. An immediate consequence of this observation is
that if $d^{*}$ is the conjugate sequence of $d$, then,

$\sum\limits_{i=1}^{n}d_{i}=\sum\limits_{i=1}^{r}d_{i}^{*}$

If $d$ represents the degree sequence of a graph, then the number of boxes in
the $i^{th}$ row of the Young diagram is the degree of vertex $i$, while the
number of boxes in the $i^{th}$ row of the Young diagram of the transpose is
the number of vertices with degree at least $i$. The trace of a Young diagram
$tr(Y(d))$ is $tr(Y(d))=|\\{i:d_{i}\geq i\\}|=tr(Y(d^{*}))$, which is the
length of “diagonal” of the Young diagram for $d$ (or $d^{*}$).

The degree sequence is a graph invariant, so two isomorphic graphs have the
same degree sequence. In general, the degree sequence does not uniquely
determine a graph, that is, two non-isomorphic graphs can have the same degree
sequence. However, for threshold graphs, we have the following result.

######  Proposition 4.1 ([15]).

Let $\Gamma_{1}$ and $\Gamma_{2}$ be two threshold graphs and let
$\pi_{1}(\Gamma_{1})$ and $\pi_{2}(\Gamma_{2})$ be degree sequences of
$\Gamma_{1}$ and $\Gamma_{2}$ respectively. If
$\pi_{1}(\Gamma_{1})=\pi_{2}(\Gamma_{2})$, then $\Gamma_{1}\cong\Gamma_{2}$.

The Laplacian spectrum of threshold graphs $\Gamma$, which we denote by $\ell-
spec(\Gamma)$, have been studied in [9, 14]. In [9], the formulas for the
Laplacian spectrum, the Laplacian polynomial, and the number of spanning trees
of a threshold graph are given. It is shown that the degree sequence of a
threshold graph and the sequence of eigenvalues of its Laplacian matrix are
“almost the same” and on this basis, formulas are given to express the
Laplacian polynomial and the number of spanning trees of a threshold graph in
terms of its degree sequence.

The following is the fascinating result regarding the Laplacian eigenvalues of
the graph realized by a finite abelian $p$-group of rank $1$.

######  Theorem 9.

[13]. Let $\Gamma(G)$ be the graph realized by a finite abelian $p$-group of
the type $G=\mathbb{Z}/p^{k}\mathbb{Z}$. Then the representatives
$0,1,p,p^{2},\cdots,p^{k-1}$ (with multiplicities) of orbits
$\\{\mathcal{O}_{k,p^{k}}\\}\cup\\{\mathcal{O}_{k,p^{i}}:0\leq i\leq k-1\\}$
of symmetric action on $G$ are the Laplacian eigenvalues of $\Gamma(G)$, that
is, $\ell-spec(\Gamma(G))=\\{0,1,p,p^{2},\cdots,p^{k-1},p^{k}\\}$.

######  Definition 10.

Let $\pi_{1},\pi_{2},\cdots,\pi_{n}\in\mathbb{Z}_{>0}$ and
$\pi_{1}^{\bullet},\pi_{2}^{\bullet},\cdots,\pi_{n}^{\bullet}\in\mathbb{Z}_{>0}$
be some partitions of $n\in\mathbb{Z}_{>0}$. A sequence (partition) of
eigenvalues $\pi=(\pi_{1},\pi_{2},\cdots,\pi_{n})$ of a graph $\Gamma$ is said
to be a threshold eigenvalues sequence (partition) if
$\pi_{i}=\pi_{i}^{\bullet}+1$ for all $i$ with $1\leq i\leq tr(Y(\pi))$.

Just for the convenience we refer the Laplacian eigenvalues as eigenvalues.
The sequence of representatives of orbits (or eigenvalues of
$\Gamma(\mathbb{Z}/p^{k}\mathbb{Z})$) of a symmetric action on a group
$\mathbb{Z}/p^{k}\mathbb{Z}$ obtained in Theorem (9) represents transpose of a
young diagram $Y(d)$, where $d$ is the degree sequence of the graph realized
by $\mathbb{Z}/p^{k}\mathbb{Z}$.

For a group $G=\mathbb{Z}/2^{4}\mathbb{Z}$ be a group, the degree sequence
$\sigma$ of $\Gamma(G)$ is,

$\sigma=\pi^{\bullet}=(15,7,3,3,2,2,2,2,1,1,1,1,1,1,1,1).$

The conjugate sequence of $\sigma$ is,

$\sigma^{*}=\pi=(2^{4},2^{3},2^{2},2,2,2,2,1,1,1,1,1,1,1,1).$

A partition $\pi$ of eigenvalues of $\Gamma(G)$ is a threshold eigenvalues
partition, since
$\sum\limits_{i=1}^{3}\pi_{i}=\sum\limits_{i=1}^{3}\pi_{i}^{\bullet}+1$. Note
that $tr(Y(\pi))=3$, the three blocks in $Y(\sigma^{*})=Y(\pi)$ are shown as
$t_{11},t_{22},t_{33}$ before the darkened column in Figure (3) below.

Figure 3: $Y(\pi)$

Thus from above discussion we assert that a partition $\pi$ of eigenvalues is
a threshold eigenvalues partition if and only if $Y(\pi)$ can be decomposed
into an $tr(Y(\pi))\times tr(Y(\pi))$ array of blocks in the upper left-hand
corner called the trace square in $Y(\pi)$. A column of $tr(Y(\pi))$ blocks
placed immediately on the right hand side of trace square, darkened in Figure
(3), and a piece of blocks on the right hand side of column $tr(Y(\pi))+1$ is
the transpose of the piece which is below the trace square.

If $a=(a_{1},a_{2},\cdots,a_{r})$ and $b=(b_{1},b_{2},\cdots,b_{s})$ are non-
increasing sequences of real numbers. Then $b$ weakly majorizes $a$, written
as $b\succeq a$, if $r\geq s$,

$\sum\limits_{i=1}^{k}b_{i}\geq\sum\limits_{i=1}^{k}a_{i},$ (1)

where $1\leq k\leq s$, and

$\sum\limits_{i=1}^{r}b_{i}\geq\sum\limits_{i=1}^{s}a_{i}.$ (2)

If $b$ weakly majorizes $a$ and equality holds in (2), then $b$ majorizes $a$,
written as $b\succ a$.

We present an example which illustrates that the threshold eigenvalues
partition of some graph realized by a finite abelian $p$-group $G_{1}$
majorizes the degree partition of the graph realized by some other finite
abelian $p$-group $G_{2}$.

Let $G_{1}=\mathbb{Z}/2^{3}\mathbb{Z}$ and $G_{1}=\mathbb{Z}/3^{2}\mathbb{Z}$
be two groups. The degree partitions $\pi_{1}^{\bullet}$ and $\pi_{2}$ of
graphs $\Gamma(G_{1})$ and $\Gamma(G_{2})$ are listed below as,

$\pi_{1}^{\bullet}=(7,3,2,2,1,1,1,1),$

$\pi_{2}=(8,2,2,1,1,1,1,1,1).$

The partitions $\pi_{1}^{\bullet},\pi_{2}\in\mathcal{P}(18)$, where
$\mathcal{P}(18)$ is the set of all partitions of $18$. The partition
$\pi_{1}=(8,4,2,1,1,1,1)$ is the threshold eigenvalues partition of
$\Gamma(G_{1})$. The Young diagrams of partitions $\pi_{1}$ and $\pi_{2}$ are
shown in Figure (4).

Figure 4: Young diagrams of $\pi_{1}$ and $\pi_{2}$

Let $\pi^{\bullet}$ and $\sigma$ be two degree sequences of graphs realized by
finite abelian $p$-groups of rank $1$ such that $\pi^{\bullet},\sigma\vdash
m$, where $m\in\mathbb{Z}_{>0}$. Then $\pi\succ\sigma$ if and only if $Y(\pi)$
can be obtained from $Y(\sigma)$ by moving blocks of the highest row in
$Y(\sigma)$ to lower numbered rows. Thus majorization induces a partial order
on sets $\\{Y(\pi^{\bullet}):\pi^{\bullet}\leavevmode\nobreak\
is\leavevmode\nobreak\ a\leavevmode\nobreak\ degree\leavevmode\nobreak\
sequence\leavevmode\nobreak\ of\leavevmode\nobreak\ some\leavevmode\nobreak\
graph\leavevmode\nobreak\ re$ $alized\leavevmode\nobreak\
by\leavevmode\nobreak\ a\leavevmode\nobreak\ p-group\leavevmode\nobreak\
of\leavevmode\nobreak\ rank\leavevmode\nobreak\ 1\\}$ and
$\\{Y(\pi^{\bullet}):\pi^{\bullet}\vdash n,n\in\mathbb{Z}_{>0}\\}$.

######  Corollary 11.

If $\pi,\sigma\in\mathcal{P}(n)$, $n\in\mathbb{Z}_{>0}$, then $\pi\succ\sigma$
if and only if $Y(\pi)$ can be obtained from $Y(\sigma)$ by moving blocks of
the highest row in $Y(\sigma)$ to lower numbered rows.

######  Theorem 12.

Let $\mathcal{T}_{p_{1}\cdots p_{n}}$ be the collection of all graphs realised
by all sequences of finite abelian $p_{r}$-groups, where $1\leq r\leq n$. If
$\pi$ is a threshold eigenvalues partition, then upto isomorphism, there is
exactly one finite abelian $p_{r}$-group $G$ of rank $1$ such that $\ell-
spec(\Gamma(G))\setminus\\{0\\}=\pi$.

###### Proof.

Let
$\left(\Gamma(\mathbb{Z}/p_{r}^{\lambda_{r,1}}\mathbb{Z}),\Gamma(\mathbb{Z}/p_{r}^{\lambda_{r,2}}\mathbb{Z}),\cdots,\Gamma(\mathbb{Z}/p_{r}^{\lambda_{r,n}}\mathbb{Z})\right)\in\mathcal{T}_{p_{1}\cdots
p_{n}}$ be a sequence of graphs realized by a sequence of finite abelian
$p_{r}$-groups
$\left(\mathbb{Z}/p_{r}^{\lambda_{r,1}}\mathbb{Z},\mathbb{Z}/p_{r}^{\lambda_{r,2}}\mathbb{Z},\cdots,\mathbb{Z}/p_{r}^{\lambda_{r,n}}\mathbb{Z}\right)$.

Let $\pi$ be a threshold eigenvalues partition of some graph of the sequence.
Without loss of generality let it be the graph realised by a finite abelian
$p_{r}$-group $\mathbb{Z}/p_{r}^{\lambda_{r,r}}\mathbb{Z}$. The partition
$\pi$ is represented by Young diagram $Y(\pi)$ and the Young diagram for the
abelian $p_{r}$-group of type $\mathbb{Z}/p_{r}^{\lambda_{r,r-1}}\mathbb{Z}$
can be obtained from $Y(\pi)$ by removing some blocks in rows and columns of
$Y(\pi)$. The proof now follows by induction on terms of the sequence of
graphs. ∎

For $1\leq i\leq j\leq n$, let $G$ be a finite abelian $p_{i}$-group of rank
$1$ and $H$ be a finite abelian $p_{j}$-group of the same rank. Moreover, let
$\Gamma(G)$ and $\Gamma(H)$ be two graphs realized by $G$ and $H$. We define a
partial order “$\leq$” on $\mathcal{T}_{p_{1}\cdots p_{n}}$. Graphs
$\Gamma(G),\Gamma(H)\in\mathcal{T}_{p_{1}\cdots p_{n}}$ are related as
$\Gamma(G)\leq\Gamma(H)$ if and only if $\Gamma(H)$ contains a subraph
isomorphic to $\Gamma(G)$, that is if and only if $\Gamma(G)$ can be obtained
from $\Gamma(H)$ by “deletion of vertices”.

The relation “degeneration” on the set $\mathcal{T}_{p_{1}\cdots p_{n}}$
descends to a partial order on $\mathcal{T}_{p_{1}\cdots p_{n}}$ and two
graphs $\Gamma(G)$, $\Gamma(H)$ are related if $\Gamma(G)$ degenerates to
$\Gamma(H)$. It is not hard to verify that the partial orders “$\leq$” and
“degeneration” are equivalent on $\mathcal{T}_{p_{1}\cdots p_{n}}$, since by
“deletion of vertices” in $\Gamma(H)$ we get the homomorphic image of
$\Gamma(G)$ in $\Gamma(H)$ and if $\Gamma(G)$ degenerates to $\Gamma(H)$, then
$\Gamma(G)$ can be obtained from $\Gamma(H)$ by “deletion of vertices”.

Recall that a poset $P$ is locally finite if the interval $[x,z]=\\{y\in
P:x\leq y\leq z\\}$ is finite for all $x,z\in P$. If $x,z\in P$ and
$[x,z]=\\{x,z\\}$, then $z$ covers $x$. A Hasse diagram of $P$ is a graph
whose vertices are the elements of $P$, whose edges are the cover relations,
and such that z is drawn “above” x whenever $x<z$.

A lattice is a poset $P$ in which every pair of elements $x,y\in P$ has a
least upper bound (or join), $x\vee y\in P$, and a greatest lower bound (or
meet), $x\wedge y\in P$. Lattice $P$ is distributive if $x\wedge(y\vee
z)=(x\wedge y)\vee(x\wedge z)$ and $x\vee(y\wedge z)=(x\vee y)\wedge(x\vee z)$
for all $x,y,z\in P$.

Let $\mathcal{Y}$ be the set of all threshold eigenvalues partitions of
members of $\mathcal{T}_{p_{1}\cdots p_{n}}$. If $\mu,\eta\in\mathcal{Y}$,
define $\mu\leq\eta$, if $Y(\mu)$ “fits in” $Y(\eta)$, that is, if
$\mu\leq\eta$, then $Y(\eta)$ is overlapped by $Y(\mu)$ or $Y(\mu)$ fits
inside $Y(\eta)$. The set $\mathcal{Y}$ with respect this partial ordering is
a locally finite distributive lattice. The unique smallest element of
$\mathcal{Y}$ is $\hat{0}=\emptyset$, the empty set.

Recall that the dual of a poset $P$ is the poset $P^{*}$ on the same set as
$P$, such that $x\leq y$ in $P^{*}$ if and only if $y\leq x$ in $P$. If $P$ is
isomorphic to $P^{*}$, then $P$ is self-dual.

######  Theorem 13.

If $\Gamma(G),\Gamma(H)\in\mathcal{T}_{p_{1}\cdots p_{n}}$, then
$\Gamma(G)\leq\Gamma(H)$ if and only $Y(\mu)$ “fits in” $Y(\eta)$, where $\mu$
and $\eta$ are threshold eigenvalues partitions of graphs $\Gamma(G)$ and
$\Gamma(H)$.

###### Proof.

If $\Gamma(G)$ is obtained from $\Gamma(H)$ by deletion of one or more
vertices, then the terms in the threshold eigenvalues partition $\mu$ are less
in number than the terms in the threshold eigenvalues partition $\eta$ of
$\Gamma(H)$. It follows that $Y(\mu)$ “fits in” $Y(\eta)$.

Conversely, suppose $Y(\mu)$ “fits in” $Y(\eta)$. The threshold eigenvalues
partitions $\mu$ and $\eta$ are obtained from degree sequences of $\Gamma(G)$
and $\Gamma(H)$. If $\Gamma(G)$ and $\Gamma(H)$ have same degree sequence,
then $\mu=\eta$. Therefore by Proposition (4.1), $\Gamma(G)\cong\Gamma(H)$.
Otherwise, $\mu\neq\eta$. Let $\Gamma(K)$ be a subgraph of $\Gamma(H)$
obtained by removing a pendant vertex from $\Gamma(H)$. Then
$Y(\eta^{\prime})$ is obtained from $Y(\eta)$ by removing a single block in
the string with number of blocks in the string equal to the largest eigenvalue
in $\eta$. It is clear that $Y(\eta^{\prime}$ “fits in” $Y(\eta)$. We continue
the process of deletion of vertices untill the resulting graph has the same
threshold eigenvalues partition as $\Gamma(G)$. Thus, it follows that
$\Gamma(H)$ contains a subgraph isomorphic to $\Gamma(H)$, that is,
$\Gamma(G)\leq\Gamma(H)$. ∎

######  Corollary 14.

The sets $\mathcal{T}_{p_{1}\cdots p_{n}}$ and $\mathcal{Y}$ are isomorphic to
each other (as posets).

###### Proof.

The bijection $\Gamma(G)\longrightarrow Y(\mu)$ is a poset isomorphism from
$\mathcal{T}_{p_{1}\cdots p_{n}}$ onto $\mathcal{Y}$, where $\mu$ is threshold
eigenvalues partition of the graph $\Gamma(G)\in\mathcal{T}_{p_{1}\cdots
p_{n}}$ realised by a finite abelian$p_{r}$-group of rank $1$. ∎

For $n\geq 1$, let $\mathcal{F}_{n}$ be the the collection of all connected
threshold graphs on $n$ vertices. We extend the partial order “$\leq$” to
$\mathcal{F}_{n}$. Two graphs $G_{1},G_{2}\in\mathcal{F}_{n}$ are related as
$G_{1}\leq G_{2}$ if and only if $G_{1}$ is isomorphic to a subgraph of
$G_{2}$. It is not difficult to verify that the poset
$\mathcal{T}_{p_{1}\cdots p_{n}}$ is an induced subposet of $\mathcal{F}_{n}$
and $\mathcal{F}_{n}$ is a self-dual distributive lattice. Moreover, if
$\mathcal{H}_{n}$ is the collection of threshold eigenvalues partitions of
members of $\mathcal{F}_{n}$, then again it is easy verify that
$\mathcal{H}_{n}$ is a poset with respect to partial order “fits in” and we
have the following observation related to posets $\mathcal{F}_{n}$ and
$\mathcal{H}_{n}$.

######  Corollary 15.

The bijection $G\longrightarrow Y(\mu)$ is a poset isomorphism from
$\mathcal{F}_{n}$ to $\mathcal{H}_{n}$, where $\mu$ is threshold eigenvalues
partition of $G\in\mathcal{F}_{n}$. In particular, $\mathcal{H}_{n}$ is self-
dual distributive lattice.

Now, we focus on sub-sequences (sub-partitions) of a threshold eigenvalues
partition. We begin by dividing $Y(\pi)$ into two disjoint pieces of blocks,
where $\pi)$ is a threshold eigenvalues partition of a graph
$\Gamma(G)\in\mathcal{T}_{p_{1}\cdots p_{n}}$. We denote by $R(Y(\pi))$ those
blocks of $Y(\pi)$ which lie on the diagonal of a trace square of $Y(\pi)$ and
to the right of diagonals. By the notation $C(Y(\pi))$, we denote those blocks
of $Y(\pi)$ that lie strictly below diagonals of a trace square, that is,
$R(Y(\pi))$ is a piece of blocks of $Y(\pi)$ on or above the diagonal and
$C(Y(\pi))$ is the piece of $Y(\pi)$ which lie strictly below the diagonal.
This process if division is illustrated as follows (Figure (5)).

Figure 5: Division of $Y(\pi)$

If we look more closely at these shifted divisions of $Y(\pi)$. Each
successive row of $R(Y(\pi))$ is shifted one block to the right. Furthermore,
$R(Y(\pi))$ corresponding to sub-partition of $\pi$ forms a strictly
decreasing sequence, that is, terms of of the sub-partition are distinct and
these sub-partitions with distinct terms are called strict threshold eigen
vales partitions. Thus, if $\pi^{\prime}=(a_{1},a_{2},\cdots,a_{n})$ is a
strict threshold eigen vales partition of a threshold eigenvalues partition
$\pi$, then there is a unique shifted division whose $i^{th}$ row contains
$a_{i}$ blocks, where $1\leq i\leq n$. It follows that there is a one to one
correspondence between the set of all threshold eigenvalue partitions of
members of $\mathcal{T}_{p_{1}\cdots p_{n}}$ and the set of all threshold
eigen vales partition. As a result, $\mathcal{Y}$ is identical to the lattice,
which we call lattice of shifted divisions.

Recall that a subset $A$ of a poset $P$ is a chain if any two elements of $A$
are comparable in $P$. A chain is called saturated if there do not exist
$x,z\in A$ and $y\in P\setminus{A}$ such that $y$ lies in between $x$ and $z$.
In a locally finite lattice, a chain $\\{x_{0},x_{1},\cdots,x_{n}\\}$ of
length $n$ is saturated if and only if $x_{i}$ covers $x_{i-1}$, where $1\leq
i\leq n$.

Since $\mathcal{T}_{p_{1}\cdots p_{n}}$ is a locally finite distributive
lattice, therefore $\mathcal{T}_{p_{1}\cdots p_{n}}$ has a unique rank
function $\Psi:\mathcal{T}_{p_{1}\cdots p_{n}}\longrightarrow\mathbb{Z}_{>0}$,
where
$\Psi\left(\Gamma(\mathbb{Z}/p_{r}^{\lambda_{r,1}}\mathbb{Z}),\cdots,\Gamma(\mathbb{Z}/p_{r}^{\lambda_{r,n}}\mathbb{Z})\right)$
is the length of any saturated chain from $\hat{0}$ to the graph realized by a
finite abelian $p_{r}$-group $\mathbb{Z}/p_{r}^{\lambda_{r,n}}\mathbb{Z}$.
Note that a finite abelian $p_{r}$-group of rank $n$,
$G_{\lambda_{r},p_{r}}=\mathbb{Z}/p_{r}^{\lambda_{r,1}}\mathbb{Z}\oplus\mathbb{Z}/p_{r}^{\lambda_{r,2}}\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/p_{r}^{\lambda_{r,n}}\mathbb{Z}$
is identified with a sequence of abelian $p_{r}$ groups of rank $1$
$\left(\mathbb{Z}/p_{r}^{\lambda_{r,1}}\mathbb{Z},\mathbb{Z}/p_{r}^{\lambda_{r,2}}\mathbb{Z},\cdots,\mathbb{Z}/p_{r}^{\lambda_{r,n}}\mathbb{Z}\right)$
which in turn is identified with a sequence of graphs
$\left(\Gamma(\mathbb{Z}/p_{r}^{\lambda_{r,1}}\mathbb{Z}),\Gamma(\mathbb{Z}/p_{r}^{\lambda_{r,2}}\mathbb{Z}),\cdots,\Gamma(\mathbb{Z}/p_{r}^{\lambda_{r,n}}\mathbb{Z})\right)$
or a sequence of a threshold partitions
$(\mu_{1},\mu_{2},\cdots,\mu_{n})\in\mathcal{Y}$. Therefore, the
correspondence of
$G_{\lambda_{r},p_{r}}=\mathbb{Z}/p_{r}^{\lambda_{r,1}}\mathbb{Z}\oplus\mathbb{Z}/p_{r}^{\lambda_{r,2}}\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/p_{r}^{\lambda_{r,n}}\mathbb{Z}$
to $(\mu_{1},\mu_{2},\cdots,\mu_{n})$ establishes that every finite abelain
$p_{r}$-group of rank $n$ can be identified with a saturated chain in
$\mathcal{T}_{p_{1}\cdots p_{n}}$ or $\mathcal{Y}$ and the rank function of
each abelian $p_{r}$-group of rank $n$ is
$\Psi(\mu_{1},\mu_{2},\cdots,\mu_{n})={\lambda_{r,n}}=$
$max\\{\lambda_{r,i}:1\leq i\leq n\\}$.

######  Remark 4.2.

Let $\Lambda_{q}$ be the set of all non-isomorphic graphs of
$\mathcal{T}_{p_{1}\cdots p_{n}}$ with equal number of edges say $q$, (graphs
realized by groups $\mathbb{Z}/2^{3}\mathbb{Z}$ and
$\mathbb{Z}/3^{2}\mathbb{Z}$ are non-isomorphic graphs with equal number of
edges). Since there is one to one correspondence between threshold eigenvalues
partitions and strict threshold eigenvalues partitions. The rank generating
function of the poset is presented in the following equation,

$\sum\limits_{q\geq 0}\kappa_{q}z^{q}=\prod\limits_{t\geq
1}(1+z^{t})=1+z+z^{2}+2z^{3}+2z^{4}+\cdots,$

where $\kappa_{q}$ is the cardinality of $\Lambda_{q}$.

The representation of a locally finite distributive lattice
$\mathcal{T}_{235}$ is illustrated in Figure (6).

Figure 6: $\mathcal{T}_{235}$

Fix a finite abelian $p_{r}$-group $G$ and
$\Gamma(G)\in\mathcal{T}_{p_{1}\cdots p_{n}}$. Let $\ell(\Gamma(G))$ be the
number of saturated chains in $\mathcal{T}_{p_{1}\cdots p_{n}}$ from $\hat{0}$
to $\Gamma(G)$.

The following result relates the number of saturated chains in
$\mathcal{T}_{p_{1}\cdots p_{n}}$ with the degree of a projective
representation of a symmetric group $\mathcal{S}_{t}$ on $t$ number of
symbols.

######  Corollary 16.

Let $\pi=(\pi_{1},\pi_{2},\cdots,\pi_{k})$ be a strict threshold eigenvalues
partition of some $\Gamma(G)\in\mathcal{T}_{p_{1}\cdots p_{n}}$. Then the
following hold,

$\ell(\Gamma(G))=\frac{t!}{\prod\limits_{i=1}^{tr(Y(\pi))}\lambda_{i}!}\prod\limits_{r<s}\frac{\lambda_{r}-\lambda_{s}}{\lambda_{r}+\lambda_{s}},$
(3)

where $\lambda_{i}=\pi_{i}-i$, $1\leq i\leq tr(Y(\pi))$ and
$\lambda=(\lambda_{1},\lambda_{2},\cdots,\lambda_{tr(Y(\pi))})$ is a partition
of some $t\in\mathbb{Z}_{>0}$.

###### Proof.

The right side of (3) represents the count of number of saturated chains from
$\hat{0}$ to $\Gamma(G)$. ∎

Note that the number of saturated chains from $\hat{0}$ to $\Gamma(G)$ in (3)
also provide a combinatorial formula for the number of finite abelian groups
of different orders.

Acknowledgement: This research project was initiated when the author visited
School of Mathematics, TIFR Mumbai, India. So, I am immensely grateful to TIFR
Mumbai for all the facilities. Moreover, I would like to thank Amitava
Bhattacharya of TIFR Mumbai for some useful discussions on this research work.

Declaration of competing interest.
There is no conflict of interest to declare.

Data Availability.
Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

## References

  * [1] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999) 434 - 447.
  * [2] R. B. Bapat, Graphs and Matrices, Springer/Hindustan Book Agency, London/New Delhi, 2010.
  * [3] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988) 208 - 226.
  * [4] Z. Bohdan, Intersection graphs of finite abelian groups, Czech. Math. Journal 25 (2) (1975) 171 - 174.
  * [5] R. Brauer, K. A. Fowler, On groups of even order, Annals of Math. 62 (2) (1955) 565 - 583.
  * [6] P. Cameron, S. Ghosh, The power graph of a finite group, Disc. Math. 311 (13) (2011) 1220 - 1222.
  * [7] V. Chvátal, P. L. Hammer, Aggregation of Inequalities in Integer Programming, Ann. Disc. Math. 1 (1977) 145 - 162.
  * [8] K. Dutta, A. Prasad, Degenerations and orbits in finite abelian groups, J. Comb. Theory, Series A 118 (2011) 1685 - 1694.
  * [9] P. L. Hammer and A. K. Kelmans, Laplacian spectra and spanning trees of threshold graphs, 65 1-3 (1996) 255 - 273.
  * [10] P. B. Henderson and Y. Zalcstein, A Graph-Theoretic Characterization of the PV class of synchronizing primitives, SIAM J. Comput. 6 (1) (1977) 88 - 108.
  * [11] M. W. Liebeck and A. Shalev, Simple Groups, Probabilistic Methods, and a Conjecture of Kantor and Lubotzky, J. Algebra 184 (1996) 31 - 57.
  * [12] N.V.R. Mahadev, U.N. Peled, Threshold Graphs and Related Topics, Ann. Disc. Math. 56 (1995).
  * [13] E. Mazumdar, Rameez Raja, Group-annihilator graphs realised by finite abelian and its properties, Graphs and Combinatorics 38 25 (2022) 25pp.
  * [14] R. Merris. Laplacian matrices of graphs: A survey, L. Algebra Appl. 197 (1994) 143 - 176.
  * [15] R. Merris. Graph Theory, John Wiley and Sons, (2011).
  * [16] F. D. Meyer and L. D. Meyer, Zero-divisor graphs of semigroups, J. Algebra 283 (2005) 190 - 198.
  * [17] G. A. Miller, Determination of all the characteristic subgroups of any abelian group, Amer. J. Math. 27 (1) (1905) 15 - 24.
  * [18] K. Mönius, Eigenvalues of zero-divisor graphs of finite commutative rings, J. Algebr. Comb. 54 (2021) 787 – 802.
  * [19] S. Pirzada, Rameez Raja, On the metric dimension of a zero-divisor graph, Commun. Algebra 45 (4) (2017) 1399 - 1408.
  * [20] Rameez Raja, Total perfect codes in graphs realized by commutative rings, Transactions of Comb. 11 (4) (2022) 295-307.
  * [21] Rameez Raja, S. Pirzada, On annihilating graphs associated with modules over commutative rings, Algebra Colloquium 29 (2) (2022) 281-296.
  * [22] M. Schwachhöfer, M. Stroppel, Finding representatives for the orbits under the automorphism group of a bounded abelian group, J. Algebra 211 (1) (1999) 225 - 239.