††thanks: _e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>(corresponding
author<EMAIL_ADDRESS>
# Continuous Phase Transition of Higher-dimensional de-Sitter Spacetime with
Non-linear Source

Yun-Zhi Du,1,2 Huai-Fan Li,1,2,∗ Li-Chun Zhang1,2 1Department of Physics,
Shanxi Datong University, Datong 037009, China
2Institute of Theoretical Physics, Shanxi Datong University, Datong, 037009,
China

###### Abstract

For the higher-dimensional dS spacetime embedded with black holes with non-
linear charges, there are two horizons with different radiation temperatures.
By introducing the interplay between two horizons this system can be regarded
as an ordinary thermodynamic system in the thermodynamic equilibrium described
by the thermodynamic quantities
($T_{eff},~{}P_{eff},~{}V,~{}S,~{}\Phi_{eff}$). In this work, our focus is on
the thermodynamic properties of phase transition for the four-dimensional dS
spacetime with different values of the charge correction $\bar{\phi}$. We find
that with the increasing of the non-linear charge correction the two horizons
get closer and closer, and the correction entropy is negative which indicates
the interaction between the two horizons stronger and stronger. Furthermore,
the heat capacity at constant pressure, isobaric expansion coefficient, and
the isothermal compression coefficient have the schottky peak at the critical
point. However, the heat capacity as constant volume for the dS spacetime is
nonzero. Finally, the dynamical properties of phase transition for this system
have investigated based on Gibbs free energy, where exists the different
behavior with that for AdS black holes.

###### pacs:

04.70.Dy 05.70.Ce

##  I Introduction

Recently people have paid more attention on the thermodynamic properties of
AdS black holes and dS spacetime with black hole in order to explore the
microstructure of black holes and the evolution of spacetime. By regarding the
cosmological constant in a $n$-dimensional AdS spacetime as pressure
($P=-\frac{\Lambda}{8\pi}=\frac{n(n-1)}{16\pi l^{2}}$), which is the
corresponding conjugate thermodynamic parameter of volume, the thermodynamic
properties of AdS black holes were investigated Caldarelli2000 ; Mann1207 ;
Cai1306 ; Wei1209 ; Wei2015 ; Banerjee1109 ; Hendi1702 ; Bhattacharya2017 ;
Zeng2017 ; Hendi1803 . When adopting the independent parameters ($P-V$) in the
charged AdS black holes, it was found that they have similar phase transition
to that of van der Waals (vdW) system Mann1207 ; Cai1306 ; Wei1209 ; Wei2015 ;
Banerjee1109 ; Zhang1502 ; Cheng1603 ; Zou1702 ; Hendi1702 ; Bhattacharya2017
; Zeng2017 ; Hendi1803 . Along with more general vdW behaviour with standard
critical exponents Dolan2014 , a broad range of other ’chemical’ black hole
behaviour was subsequently discovered, including reentrant phase transitions
Altamirano2013 , tricritical points Altamirano2014 , Carnot cycles Johnson2014
, isolated critical points Frassino2014 ; Dolan2014a , extensions to black
rings Altamirano2014a , and superfluidity Hennigar2017 .

As is well known our universe in the early period of inflation was a quasi-dS
spacetime. The cosmological constant term introduced in a dS spacetime is the
contribution of vacuum energy, which is also a form of material energy. When
regarding the cosmological constant as dark energy, our universe will evolve
into a new dS phase. In order to construct the entire history of the evolution
for our universe, we should have a clear understanding of dS spacetime.
However this is a tricky problem, since the absence of a killing vector that
is everywhere timelike outside the black hole horizon renders a good notion of
the asymptotic mass questionable. Furthermore, the presence of both the black
hole horizon and cosmological horizon that of two distinct hawking
temperatures suggests that the system does not meet the requirements of
thermodynamic equilibrium stability. These features bring certain difficulties
when we investigate the thermodynamic properties of dS spacetime. There have
been a few attempts to overcome this problem and to investigate the
thermodynamics of black holes in dS spacetime. One approach involves studying
the horizons in a dS spacetime separately, considering them as separate
thermodynamic systems characterized by their own temperature and thermodynamic
behaviours Kubiznak2016 ; Dolan2013 ; Cai2002 ; Cai2002a ; Sekiwa2006 ;
Ali2019 . Others have investigated putting the black hole inside a cavity and
imposing thermodynamic equilibrium in this closed system Carlip2003 ;
Braden1990 ; Simovic2018 , which solves the thermodynamic equilibrium problem
at the price of isolating the cosmological horizon and ignoring its
contribution to thermodynamics of the system. In addition, authors Mbarek2019
had added scalar hair to the dS black hole to ensure equilibrium between two
horizons, where only a reverse Hawking-Page transition is possible and the
total entropy is the sum of two horizons (i.e., ignoring the interaction
between two horizons). On this basis, people proposed that the total entropy
in a dS spacetime with the thermodynamic equilibrium should contain the
interaction between two horizons Guo2020 ; Zhang2019 ; Ma2020 ; Dinsmore2020 .
From this viewpoint, we will investigate the thermodynamic phase transition
for a dS spacetime with non-linear source as an ordinary thermodynamic system
in the thermodynamic equilibrium.

With the development of phase transition and microstructure for various AdS
black holes and the dS spacetime embedded black holes, people attempt to probe
the dynamic process of AdS black hole phase transition. Recently from the view
point of the Gibbs free energy, the authors in Ref. Li2020a  probed the
dynamics of switching between the coexistence black hole phases by solving the
Fokker-Planck equating with different reflection/aborption boundary conditions
and initial conditions, and calculating the mean first passage time. In this
approach, the phase transition is due to the thermal fluctuation and $G$ is
regarded an function of black hole horizon which is the order parameter of
phase transition. Subsequently, this method was applied to the HP phase
transition in Einstein gravity Li2021  and in massive gravity Li2020  and the
large/small black hole phase transition in Gauss-Bonnet gravity Wei2021 , in
Einstein gravity Yang2021  minimally coupled to nonlinear electrodynamics
Kumara2021 , and in dilaton gravity Mo2021 . However there is no works on the
dynamic phase transition for dS spacetime embedded black holes. In this paper
from the view of Gibbs free energy, we firstly attempt to exhibit the dynamic
process phase transition of the four-dimensional dS spacetime embedded the
black hole with non-linear charge.

Since the most physical systems are inherently non-linear in the nature, the
non-linear field theories are of interest to different branches of
mathematical physics. The main reason to consider the non-linear
electrodynamics (NLED) is that the structures of these theories are
considerably richer than the Maxwell field, and in special case they can
reduce to the linear Maxwell theory (LMT). Various limitations of LMT (the
self-interaction of virtual electron-positron pairs Heisenberg1936 ;
Yajima2001 ; Schwinger1951  and the radiation propagation inside specific
materials Lorenci2001 ; Lorenci2002 ; Novello2003 ; Novello2012 ) motivate
ones to consider NLED. The authors in Cavaglia2003  showed that NLED objects
can remove both of the big bang and black hole singularities. Moreover, from
astrophysical point of view, one finds that the effects of NLED become indeed
quite important in superstrongly magnetized compact objects, such as pulsars
and particular neutron stars (also the so-called magnetars and strange quark
magnetars) Cavaglia2004 . Recently the authors in Hendi2015  presented the
$n+1$-dimensional topological static black hole solutions of Einstein gravity
in presence of the mentioned NLED and checked the first law of thermodynamics.
Furthermore, they studied the stability of the solutions in both canonical and
grand canonical ensembles and analyzed the effect of the non-linear charge
correction on the thermodynamic properties of black hole. Therefore, there is
naturally a question: does a dS spacetime with the non-linear charge source
have the thermodynamic properties similar to a AdS black hole? In this work,
we regard the higher-dimensional dS spacetime with the non-linear charge
correction as an ordinary thermodynamic system by considering the interplay
between two horizons and mainly investigate the properties of phase transition
of the four-dimensional dS spacetime. And the effect of the non-linear charge
correction on the phase transition is also analyzed.

In this paper, we briefly review the thermodynamic quantities of the higher-
dimensional dS spacetime with the non-linear source in the thermodynamic
equilibrium by introducing the interplay between the black hole horizon and
cosmological horizon in Sec. II. Then the phase diagrams in $P_{eff}-V$ and
$G_{T_{eff}}-P_{eff}$ for the four-dimensional dS spacetime with different
non-linear charge corrections are presented. We have shown the behaviors of
the heat capacities nearby the critical point for this system undergoing the
phase transition in the isocratic and isobaric processes as well as the curves
of the isobaric expansion coefficient and the isothermal compression
coefficient with the radio between two horizons for the different charge
correction $\bar{\phi}$ in Sec. III. From the inspect of Gibbs free energy,
the dynamic process of phase transition is discribed in Sec. IV. Finally, a
brief summary is given in Sec. V.

##  II The higher-dimensional Topological black hole with the non-linear
source

In Refs. Zhang2019 ; Hendi2015 ; Hendi2015a ; Hendi2015b ; Zhao2021  the
authors had shown the $n+1$ dimensional action of Einstein gravity of the non-
linear source as

$\displaystyle I_{G}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\int_{M}d^{n+1}x\sqrt{-g}[R-2\Lambda+L(F)]-\frac{1}{8\pi}\int_{\partial
M}d^{n}x\sqrt{-\gamma}\Theta(\gamma),$ (1) $\displaystyle L(F)$
$\displaystyle=$ $\displaystyle-F+\alpha F^{2}+o(\alpha^{2}),$ (2)

where R and $\Lambda$ are the scalar curvature and the cosmological constant,
respectively. And $L(F)$ is the Lagrangian of non-linear source with the
Maxwell invariant $F^{2}=F^{\mu\nu}F_{\mu\nu}$, in which
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor and $A_{\mu}$ is the gauge potential. The non-
linear charge parameter $\alpha$ is small, so the effects of the non-linear
should be regarded as a perturbation. The $n+1$-dimensional topological black
hole solutions are given by

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2}_{n-1}$ (3)

with

$f(r)=k-\frac{m}{r^{n-2}}-\frac{2\Lambda
r^{2}}{n(n-1)}+\frac{2q^{2}}{(n-1)(n-2)r^{2n-4}}-\frac{4q^{4}\alpha}{(3n^{2}-7n+4)r^{4n-6}}.$
(4)

Here $m$ is an integration constant which is related to the mass of the black
hole ($M=\frac{V_{n-1}(n-1)m}{16\pi}$), and the last term in Eq. (4) indicates
the effect of the non-linearity. The asymptotical behavior of the solution is
the AdS or dS provided $\Lambda<0$ or $\Lambda>0$, and the asymptotically flat
solution is for $\Lambda=0$, and $k=1$. In the following we pay attention on
the dS black hole. The horizons of the black hole and the cosmology
($r_{+},~{}r_{c}$) are satisfied with the form $f(r_{+,c})=0$ for $\Lambda>0$.
And the radiation temperatures at two horizon surfaces were given in Refs.
Heisenberg1936 ; Hendi2015 ; Zhao2021 .

When regarding the higher-dimensional topological dS spacetime with the non-
linear source as an ordinary thermodynamic system in the thermodynamic
equilibrium, the thermodynamic quantities satisfy the first law

$\displaystyle dM=T_{eff}dS-P_{eff}dV+\Phi_{eff}dQ$ (5)

whit the definition $Q\equiv\frac{q}{4\pi}V_{n-1}$. The corresponding
thermodynamical volume and entropy read

$\displaystyle
V=V_{c}-V_{+}=\frac{V_{n-1}r_{c}^{n}}{n}(1-x^{n}),~{}~{}~{}~{}~{}S=\frac{V_{n-1}r_{c}^{n-1}}{4}F(x)$
(6)

with $x=\frac{r_{+}}{r_{c}}$, $V_{n-1}=\frac{2\pi^{n/2}}{\Gamma(n/2)}$, and

$\displaystyle
F(x)=\frac{3n-1}{2n-1}(1-x^{n})^{\frac{n-1}{n}}-\frac{n(1+x^{2n-1})+(2n-1)(1-2x^{n}-x^{2n-1})}{(2n-1)(1-x^{n})}+1+x^{n-1}=\bar{f}(x)+1+x^{n-1}.$
(7)

The effective temperature, effective potential, effective pressure, and mass
were shown as the following in Ref. Zhang2019 ; Zhao2021

$\displaystyle T_{eff}$ $\displaystyle=$ $\displaystyle\frac{B(x,q)}{4\pi
r_{c}x^{n-2}(1+x^{n+1})},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}P_{eff}=\frac{g(x,q)}{16\pi
r_{c}^{2}x^{n-2}(1+x^{n+1})},$ (8) $\displaystyle\Phi_{eff}$ $\displaystyle=$
$\displaystyle\frac{q(1-x^{2n-2})}{r_{+}^{n-2}(1-x^{n})}\left(\frac{1}{n-2}-\frac{4q^{2}\alpha(n-1)(1+x^{2n-2})}{[2(n^{2}-4)+(n-3)(n-4)]r_{+}^{2n-2}}\right),$
(9) $\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{V_{n-1}(n-1)r_{c}^{n-2}}{16\pi(1-x^{n})}\left(k(x^{n-2}-x^{n})+\frac{2q^{2}(1-x^{2n-2})}{(n-1)(n-2)r_{c}^{2n-4}x^{n-2}}-\frac{4\bar{\phi}(1-x^{2n-4})}{[2(n^{2}-4)+(n-3)(n-4)]x^{3n-4}}\right),$
(10)

where $\bar{\phi}=\frac{q^{4}\alpha}{r_{+}^{4n-6}}$,

$\displaystyle B(x,q)$ $\displaystyle=$ $\displaystyle
kx^{n-3}\left[(n-2-nx^{2})(1-x^{n})+2(n-1)x^{n}(1-x^{2})\right]$ (11)
$\displaystyle+\frac{2q^{2}x^{n-3}}{(n-1)(n-2)r_{+}^{2n-4}}\left[nx^{n}(1-x^{n-2})-(n-2)(1-x^{3n-2})\right]$
$\displaystyle+\frac{4\bar{\phi}x^{n-3}}{3n^{2}-7n+4}\left[(3n-4)x^{n}(1-x^{4n-4})-4(n-1)x^{n}+nx^{4n-4}+3n-4\right],$
$\displaystyle g(x,q)$ $\displaystyle=$ $\displaystyle
F^{\prime}\left[k(n-2)x^{n-2}(1-x^{2})-\frac{2q^{2}x^{n-2}(1-x^{2n-2})}{(n-1)r_{+}^{2n-4}}+\frac{4(3n-4)\bar{\phi}x^{n-3}(1-x^{4n-4})}{2(n^{2}-4)+(n-3)(n-4)}\right]$
(12)
$\displaystyle-\frac{4(n-1)F\bar{\phi}x^{n-3}}{2(n^{2}-4)+(n-3)(n-4)}\left[\frac{nx^{n}(1-x^{4n-4})}{(1-x^{n})}-nx^{4n-4}-(3n-4)\right]$
$\displaystyle+\frac{2q^{2}(n-1)F}{(n-1)(n-2)r_{c}^{2n-4}x^{n-1}}\left[\frac{nx^{n}(1-x^{2n-2})}{(1-x^{n})}-nx^{2n-2}-(n-2)\right]$
$\displaystyle+k(n-1)Fx^{n-3}\left[\frac{nx^{n}(1-x^{2})}{1-x^{n}}+n-2-nx^{2}\right].$

Note that the total entropy in Eq. (6) is not only the sum of entropy at the
black hole horizon and the cosmological horizon, and $\bar{f}(x)$ in the form
of $F(x)$ represents the extra contribution from the correlations of the two
horizons. We called $\bar{\phi}$ as the non-linear charge correction, which
represents the effect of the non-linear source.

##  III Phase transition of the four-dimensional dS spacetime with the non-
linear source

In the following, we will mainly focus on the thermodynamical phase transition
of the four-dimensional dS spacetime with the non-linear source. With Eq. (8)
and $n=3$, the effective temperature and the effective pressure can be
rewritten as

$\displaystyle 0$
$\displaystyle=T_{eff}f_{1}(x)-\frac{f_{2}(x)}{r_{+}}+\frac{q^{2}f_{3}(x)}{r_{+}^{3}},$
(13) $\displaystyle 0$
$\displaystyle=P_{eff}f_{4}(x)r_{+}^{4}+f_{5}(x)r_{+}^{2}+q^{2}f_{6}(x)$ (14)

with

$\displaystyle f_{1}(x)\\!$ $\displaystyle=$
$\displaystyle\\!\frac{4\pi(1+x^{4})}{1-x},~{}~{}f_{3}(x)=(1+x+x^{2})(1+x^{4})-2x^{3},~{}~{}f_{4}(x)=\frac{8\pi(1+x^{4})}{x(1-x)},$
$\displaystyle f_{2}(x)\\!$ $\displaystyle=$
$\displaystyle\\!\frac{2\bar{\phi}}{5}\left[5(1+x+x^{2})(1+x^{8})+2x^{3}(1+x+x^{2}+x^{3}+x^{4})\right]$
$\displaystyle+k\left[(1-3x^{2})(1+x+x^{2})+4x^{3}(1+x)\right],$
$\displaystyle f_{5}(x)\\!$ $\displaystyle=$
$\displaystyle\\!kx(1+x)F^{\prime}/2-\frac{k(1+2x)F}{1+x+x^{2}}-\bar{\phi}F^{\prime}(1+x)(1+x^{2})(1+x^{4})$
$\displaystyle-\frac{2\bar{\phi}F(5+10x+15x^{2}+12x^{3}+9x^{4}+6x^{5}+3x^{6})}{5(1+x+x^{2})},$
$\displaystyle f_{6}(x)\\!$ $\displaystyle=$
$\displaystyle\\!\frac{(1+2x+3x^{2})F}{1+x+x^{2}}+x(1+x)(1+x^{2})F^{\prime}/2.$

The volume and entropy become

$\displaystyle
V=\frac{V_{2}r_{+}^{3}(1-x^{3})}{3x^{2}},~{}~{}~{}~{}~{}~{}S=\frac{V_{2}r_{+}^{2}F(x)}{4x^{2}}$
(15)

with $V_{2}=\frac{2\pi^{3/2}}{\Gamma(3/2)}$.

When the system undergoing an isothermal and isobaric processes with Eqs. (13)
and (14), the physical horizon radius $r_{+}$ satisfies the following
expressions, respectively

$\displaystyle r_{+}$ $\displaystyle=$ $\displaystyle
r_{t}=\frac{1}{2\cos(\theta+4\pi/3)}\sqrt{\frac{3q^{2}f_{3}}{f_{2}}},~{}~{}~{}\theta=\frac{1}{3}\arccos\left(-\frac{qT_{eff}f_{1}}{2f_{2}^{2}}\sqrt{27f_{2}f_{3}}\right),$
(16) $\displaystyle r_{+}$ $\displaystyle=$ $\displaystyle
r_{p}=\sqrt{\frac{-f_{5}+\sqrt{f_{5}^{2}-4q^{2}P_{eff}f_{4}f_{6}}}{2P_{eff}f_{4}}}.$
(17)

The critical point is determined by the following expressions

$\displaystyle\left(\frac{\partial P_{eff}}{\partial
V}\right)_{T_{eff},q}=\left(\frac{\partial^{2}P_{eff}}{\partial
V^{2}}\right)_{T_{eff},q}=0.$ (18)

Table 1: Critical point with different charge correction $\bar{\phi}$ for $k=1,~{}q=1$ charge corrections | $\bar{\phi}=0$ | $\bar{\phi}=10^{-5}$ | $\bar{\phi}=0.001$ | $\bar{\phi}=0.002$ | $\bar{\phi}=0.005$ | $\bar{\phi}=0.01$
---|---|---|---|---|---|---
$x_{c}=\frac{r_{+}}{r_{c}}$ | 0.65646 | 0.656476 | 0.65797735 | 0.659514 | 0.664132 | 0.6718755
$r^{c}_{+}$ | 2.6438 | 2.64382 | 2.64723 | 2.65037 | 2.66029 | 2.6783
$r^{c}_{c}$ | 4.02735 | 4.02728 | 4.02328 | 4.01867 | 4.00566 | 3.98631
$V^{c}$ | 196.214 | 196.199 | 195.083 | 193.87 | 190.359 | 184.863
$S^{c}$ | 68.2089 | 68.2078 | 68.1809 | 68.137 | 68.0382 | 67.9733
$T_{eff}^{c}$ | 0.00863395 | 0.00863388 | 0.00862669 | 0.00861906 | 0.00859394 | 0.00854405
$P_{eff}^{c}$ | 0.000583686 | 0.000583694 | 0.000584461 | 0.000585188 | 0.000587074 | 0.000589151

By solving the above equations, we can obtain the thermodynamical critical
quantities with different values of the charge correction $\bar{\phi}$ for
$k=1,~{}q=1$ in Tab. 1. It is obvious from the Tab. 1 that the phase
transition will emerge with the condition $x=x_{c}$ and $r_{+}=r_{+}^{c}$ for
fixed non-linear charge correction. The critical radio, radius (of the dS
black hole horizon), and effective pressure are both increasing monotonically
with the increase of the charge correction $\bar{\phi}$, while the critical
radius (for the cosmological horizon), volume, effective temperature, and
entropy are decreasing monotonically. That means that with the increasing of
non-linear charge correction, the two horizons are getting closer and closer
to each other through attraction and their interaction is becoming stronger
and stronger, the correlation entropy $\bar{f}(x)$ between the two horizons is
negative.

Taking the conjugate quantities $(P_{eff}-V)$ and considering Eqs. (6), (13),
(14), and (16), we exhibit the phase transition diagrams undergoing the
isothermal processes with different non-linear charge corrections in Fig. 1.
For $T_{eff}<T_{eff}^{c}$, the phase transition of the four-dimensional dS
spacetime with the non-linear charge correction will emerge, while there is no
phase transition for $T_{eff}>T_{eff}^{c}$.

(a) $\bar{\phi}=10^{-5}$ (b) $T_{eff}=0.008591$

Figure 1: $P_{eff}-V$ with different non-linear corrections and different
effective temperatures for the parameters $k=1,~{}q=1$.

Gibbs free energy is an important thermodynamic quantity to investigate the
phase transition in addition to the equal area law. For the first-order phase
transition, it exhibits a swallow tail behavior. At the second-order phase
transition point, the Gibbs free energy is continuous but not smooth.
Moreover, it also corresponds to the thermal dynamic phase transition of a
black hole. From Eq. (10) and $n=3$, we have the mass of the four-dimensional
dS spacetime as

$\displaystyle
M=\frac{V_{2}r_{+}(1-x^{2})}{8\pi(1-x^{3})}\left[k+\frac{q^{2}(1+x^{2})}{r_{+}^{2}}-\frac{2\bar{\phi}}{5}\right].$
(19)

And the Gibbs free energy reads

$\displaystyle G(r_{+},x)=M-T_{eff}S+P_{eff}V.$ (20)

Figure 2: $G_{Teff}-P_{eff}$ with the non-linear correction
$\bar{\phi}=10^{-5}$ and $k=1,~{}q=1$. The effective temperature is set to
$1.001T_{eff}^{c}$ (black lines), $T_{eff}^{c}$ (blue lines),
$0.997T_{eff}^{c}$ (green lines), $0.995T_{eff}^{c}$ (red lines),
respectively.

For this system undergoing the isothermal processes, considering Eqs. (6),
(13), (14), (16), and (19), we present the Gibbs free energy nearby the
critical points with different non-linear charge corrections in Fig. 2. For
low temperature, there appears the swallow tail behavior. At the effective
critical temperature, this characterized behavior disappears. When beyond the
critical point, Gibbs free energy is an increasing function of the pressure,
and no phase transition emerges. Reading out the self-cross point, one can
obtain the phase transition effective temperature and pressure.

The heat capacities at constant pressure and at constant volume are
respectively

$\displaystyle C_{V}$ $\displaystyle=$ $\displaystyle
T_{eff}\left(\frac{\partial S}{\partial
T_{eff}}\right)_{V}=T_{eff}\left(\frac{\frac{\partial S}{\partial
r_{+}}\frac{\partial V}{\partial x}-\frac{\partial S}{\partial
x}\frac{\partial V}{\partial r_{+}}}{\frac{\partial T_{eff}}{\partial
r_{+}}\frac{\partial V}{\partial x}-\frac{\partial T_{eff}}{\partial
x}\frac{\partial V}{\partial r_{+}}}\right),$ (21) $\displaystyle C_{P_{eff}}$
$\displaystyle=$ $\displaystyle T_{eff}\left(\frac{\partial S}{\partial
T_{eff}}\right)_{P_{eff}}=T_{eff}\left(\frac{\frac{\partial S}{\partial
r_{+}}\frac{\partial{P_{eff}}}{\partial x}-\frac{\partial S}{\partial
x}\frac{\partial{P_{eff}}}{\partial r_{+}}}{\frac{\partial T_{eff}}{\partial
r_{+}}\frac{\partial{P_{eff}}}{\partial x}-\frac{\partial T_{eff}}{\partial
x}\frac{\partial{P_{eff}}}{\partial r_{+}}}\right).$ (22)

The isobaric expansion coefficient and the isothermal compression coefficient
have the following forms

$\displaystyle\alpha_{P_{eff}}$ $\displaystyle=$
$\displaystyle\frac{1}{V}\left(\frac{\partial V}{\partial
T_{eff}}\right)_{P_{eff}}=\frac{1}{V}\left(\frac{\frac{\partial V}{\partial
r_{+}}\frac{\partial P_{eff}}{\partial x}-\frac{\partial V}{\partial
x}\frac{\partial P_{eff}}{\partial r_{+}}}{\frac{\partial T_{eff}}{\partial
r_{+}}\frac{\partial P_{eff}}{\partial x}-\frac{\partial T_{eff}}{\partial
x}\frac{\partial P_{eff}}{\partial r_{+}}}\right),$ (23)
$\displaystyle\kappa_{T_{eff}}$ $\displaystyle=$
$\displaystyle-\frac{1}{V}\left(\frac{\partial V}{\partial
P_{eff}}\right)_{T_{eff}}=-\frac{1}{V}\left(\frac{\frac{\partial V}{\partial
r_{+}}\frac{\partial T_{eff}}{\partial x}-\frac{\partial V}{\partial
x}\frac{\partial T_{eff}}{\partial r_{+}}}{\frac{\partial P_{eff}}{\partial
r_{+}}\frac{\partial T_{eff}}{\partial x}-\frac{\partial P_{eff}}{\partial
x}\frac{\partial T_{eff}}{\partial r_{+}}}\right).$ (24)

With Eqs. (6), (13), (14), (16), and (17), we have exhibited the pictures of
$C_{V}-T_{eff},~{}C_{P_{eff}}-T_{eff},~{}\alpha_{P_{eff}}-x,$ and
$\kappa_{T_{eff}}-x$ in Figs. 3 and 4.

(a) $C_{V}-T_{eff}$ (b) $C_{P_{eff}}-T_{eff}$

Figure 3: The heat capacities as the functions of the effective temperature
$T_{eff}$ nearby the critical point with different non-linear correction for
the parameters $k=1,~{}q=1$.

(a) $\alpha_{P_{eff}}-x$ (b) $\kappa_{T_{eff}}-x$

Figure 4: The isobaric expansion coefficient and the isothermal compression
coefficient as the functions of the radio $x$ ($x=\frac{r_{+}}{r_{c}}$) nearby
the critical point with the different charge corrections and the parameters
$k=1,~{}q=1$.

From the Figs. 3 and 4, for the dS spacetime with the certain non-linear
charge correction $\bar{\phi}$ the heat capacity at constant pressure, the
isobaric expansion coefficient, and the isothermal compression coefficient at
the critical point are all of the schottky peak. That is consistent with the
conclusions for an ordinary thermodynamic system in the thermodynamic
equilibrium. Furthermore, the schottky peak in $C_{P_{eff}}-T_{eff}$ moves
toward the left with the increasing of $\bar{\phi}$, i.e., the effective
critical temperature becomes smaller with the nonlinear charge correction. In
diagrams $\alpha_{P_{eff}}-x$ and $\kappa_{T_{eff}}-x$, it moves toward the
right, which indicates the critical radio
$x_{c}~{}(x_{c}=\frac{r_{+}^{c}}{r_{c}^{c}})$ becomes bigger. Those properties
are fully consistent with what we had shown in Tab. 1. Note that it is very
interesting that the heat capacity at constant volume is nonzero, which is
completely contrary to that in a AdS black hole. That can be regarded a
difference between dS spacetime and AdS black hole. In addition, the heat
capacity at constant volume decreases with the increasing of the non-linear
charge correction.

##  IV Dynamic Properties of Thermodynamic Phase Transition

Recently, authors in Refs. Li2020a  proposed that Gibbs free energy is related
with the thermal dynamic phase transition of RN-AdS black hole. Subsequently
this idea was applied to various AdS black holes Li2020 ; Li2021 ; Wei2021 ;
Yang2021 ; Kumara2021 ; Mo2021 . Based on this, in the following we will
investigate the thermal dynamic phase transition for this system from the view
of Gibs free energy.

The picture of Gibbs free energy at phase transition point of
$P_{0}=0.0005749$ and $T_{0}=0.995Tc$ is exhibited in Fig. 5. From this
picture, we can see that the Gibbs free energy displays the double-well
behavior. Namely there are two local minimum (located at
$r_{s}=2.447,~{}r_{l}=2.884$) which are corresponding to the stable
small/large dS black hole with positive heat capacity. The local maximum
located at $r_{m}=2.659$ stands for the unstable intermediate-dS black hole
state with negative heat capacity and acts as a barrier between the stable
small and large dS black holes. Furthermore, the depths of two local minimum
are the same. It indicates that the phase transition will occur at the case of
the same depth for two wells from the view of Gibbs free energy. At this
issue, we expect that the reentrant phase transition or triple point maybe
correspond to more wells of Gibbs free energy.

Figure 5: Gibbs free energy as the function of $r_{+}$ with the non-linear
charge correction $\bar{\phi}=10^{-5}$ and the parameters $k=1,~{}q=1$.

###  IV.1 Fokker-Planck Equation and Probabilistic Evolution

Recently authors proposed the stochastic dynamic process of AdS black hole
phase transition can be studied by the associated probabilistic Fokker-Planch
equation on Gibbs free energy landscape Li2020a ; Li2020 ; Li2021 ; Wei2021 ;
Yang2021 ; Kumara2021 ; Mo2021 , which is an equation of motion governing the
distribution function of fluctuating macroscopic variables. For a AdS black
hole thermodynamic system, the horizon $r_{+}$ is the order parameter and it
can be regarded as a stochastic fluctuating variable during phase transition.
Based on this, by regarding the dS black hole horizon as the order parameter
and a stochastic fluctuating variable for this dS spacetime we will exhibit
the dynamical process of phase transition in the canonical ensemble under the
thermal fluctuations. Note that the canonical ensemble is consisted of a
series of dS spacetime embedded with arbitrary dS black hole horizons.

The probability distribution of these dS spacetimes $\rho(t,r_{+})$ satisfies
the Fokker-Planck equation on Gibbs free energy

$\displaystyle\frac{\partial\rho(t,r_{+})}{\partial
t}=D\frac{\partial}{\partial r_{+}}\left(e^{-\beta
G(r_{+},x)}\frac{\partial}{\partial r_{+}}\left[e^{\beta
G(r_{+},x)}\rho(t,r_{+})\right]\right),$ (25)

where $\beta=1/kT$, $D=kT/\xi$ is the diffusion coefficient with $k$ being the
Boltzman constant and $\xi$ being dissipation coefficient. Without loss of
generality, we set $k=\xi=1$. Note that although Gibbs free energy is a
function of $r_{+}$ and $x$, the probability distribution is still selected as
a function of $r_{+}$ and $t$ by substituting the relationship between $x$ and
$r_{+}$ in certain way. In order to solve the above equation, two types of
boundary ($r_{+}=r_{0}$) condition should be imposed. One is the reflection
boundary condition, which preserves the normalization of the probability
distribution. The other is the absorption boundary condition.

In this system the location of left boundaries should be smaller than $r_{s}$,
and the right one is bigger than $r_{l}$. Since the effective temperature of
the system with the effective pressure $P_{0}=0.0005749$ has a minimum value
$0.0081799$, there exists the minimum value of $r_{+}$: $r_{min}=1.90074$. We
can regard $r_{min}$ as the left boundary $r_{lb}$, and set the right one
$r_{rb}$ as $4$. The reflection boundary condition means the probability
current vanishes at the left and right boundaries:

$\displaystyle j(t,r_{0})=-T_{E}e^{-G_{L}/T_{E}}\frac{\partial}{\partial
r_{+}}\left(e^{G_{L}/T_{E}}\rho(t,r_{+})\right)\mid_{r_{+}=r_{0}}=0$ (26)

And the absorption one means the vanishing probability distribution function
at the boundary: $\rho(t,r_{0})=0$. The adoption of boundary condition is
determined by the considering physical problem.

The initial condition is chosen as a Gaussian wave packet located at $r_{i}$:

$\displaystyle\rho(0,r_{+})=\frac{1}{\sqrt{\pi}}e^{-\frac{(r_{+}-r_{i})^{2}}{a^{2}}}.$
(27)

Here $a$ is a constant which determines the initial width of Gaussian wave
packet and it does not influence the final result. Since we mainly consider
the thermal dynamic phase transition, $r_{i}$ can be set to $r_{s}$ or
$r_{l}$. It means this thermal system is initially at the dS spacetime
embedded with the small or large dS black hole.

The time evolution of the probability distribution are shown in Figs. 6 and 7.
As $t=0$ the Gaussian wave packets locate at the dS spacetime state with the
large dS black hole and with the small dS black hole when $T_{eff}=0.995T_{c}$
and $a=0.1$, respectively. And they are both decreasing with increasing time
$t$ until tending to a certain constant. However, at the same time the peaks
of $\rho(t,r_{+})$ at $r_{+}=r_{s}$ (Fig. 6(b)) and $r_{+}=r_{l}$ (Fig. 7(b))
are increasing from zero to the same constant. This indicates that the system
embedded with large dS black hole tends to the phase with small dS black hole
as shown Fig. 6(b), while it embedded with small dS black hole is tending to
the one with large dS black hole as shown Fig. 7(b). It should be noticed that
the peak behaviors of $\rho(t,r_{+})$ at $r_{+}=r_{s}$ (Fig. 6(b)) and
$r_{+}=r_{l}$ (Fig. 7(b)) are not monotonous increasing in a short time near
the initial time comparing with a AdS black hole, which will lead to a
conjecture that the microstructure of dS spacetime embedded black holes may be
different from that in AdS black holes. Finally the system reaches a
coexistence stationary state at a short time. This is consistent with what has
shown in $G-r_{+}$, i.e., Gibbs free energy of the dS spacetime with large
black hole and small black hole have the same depth in the double well.

(a) $r_{i}=r_{l}$

(b) $r_{i}=r_{l}$

Figure 6: $\rho-t-r_{+}$ with the parameters
$k=1,~{}q=1,~{}\bar{\phi}=10^{-5}$. The effective temperature is set to
$0.995T_{c}$.

(a) $r_{i}=r_{s}$

(b) $r_{i}=r_{s}$

Figure 7: $\rho-t$ with the parameters $k=1,~{}q=1,~{}\bar{\phi}=10^{-5}$. The
effective temperature is set to $0.995T_{c}$.

###  IV.2 First Passage Time

In general, the important quantity in the dynamical process of phase
transition is characterized by the first passage time, which is defined as the
mean value of the first passage time that the dS spacetime with a stable large
black hole or a stable small black hole scape to the one with a unstable
intermediate black hole (i.e., from the one well to the barrier of Gibbs free
energy).

Supposing there is a perfect absorber in a stable state, if the system makes
the first passage under the thermal fluctuation, the system will leave this
state. We can define $\Sigma$ to be the sum probability of the dynamical
process within the first passage time as

$\displaystyle\Sigma=\int^{r_{m}}_{r_{min}}\rho(t,r_{+})dr_{+},~{}~{}~{}~{}~{}~{}or~{}~{}~{}~{}~{}\Sigma=-\int^{r_{m}}_{r_{rb}}\rho(t,r_{+})dr_{+},$
(28)

where $r_{m}$, $r_{min}$, $r_{rb}$ are the intermediate, minimum, and right
boundary of charge dS black hole horizons. At a long time, the probability of
this system becomes zero, i.e., $\Sigma(t,r_{l})\mid_{t\rightarrow\infty}=0$
or $\Sigma(t,r_{s})\mid_{t\rightarrow\infty}=0$. As claimed, the first passage
time is a random variable because the dynamical process of phase transition is
caused by thermal fluctuation. Hence, we denote the distribution of the first
passage time by $F_{p}$, which reads

$\displaystyle F_{p}=-\frac{d\Sigma}{dt}.$ (29)

It is obviously that $F_{p}dt$ indicates the probability of the dS spacetime
with the stable large or small black hole passing through the one with
unstable intermediate black hole for the first passage time in the time
interval ($t,~{}t+dt$). Considering Eqs. (25) and (28), the distribution of
the first passage time $F_{p}$ becomes Li2020a

$\displaystyle F_{p}=-D\frac{\partial\rho(t,r_{+})}{\partial
r}\mid_{r_{m}},~{}~{}~{}~{}~{}or~{}~{}~{}~{}F_{p}=D\frac{\partial\rho(t,r_{+})}{\partial
r}\mid_{r_{m}}$ (30)

Here the absorbing and reflecting boundary conditions of the Fokker-Planck
equation are imposed at $r_{m}$ and the other end ($r_{min}$ or $r_{rb}$).
Note that the normalisation of the probability distribution is not preserved.

(a) $r_{i}=r_{l}$

(b) $r_{i}=r_{l}$

Figure 8: The pictures of $\Sigma-t$ and $F_{p}-t$ with the parameters
$k=1,~{}q=1,~{}\bar{\phi}=10^{-5}$. The effective temperature is set to
$0.995T_{c}$ (blue lines) and $0.997T_{c}$ (dashed red lines), respectively.

By solving the Fokker-Planck equation (25) with different phase transition
temperatures ($T_{eff}=0.995T_{c}$ and $T_{eff}=0.997T_{c}$) and substituting
them into Eqs. (28) and (29), the numerical results are displayed in Fig 8 for
the initial condition $r_{i}=r_{l}$. It is obviously clear that in Fig. 8(a),
$\Sigma$ decays very fast in a short time. Furthermore further increasing the
effective temperature the probability of $\Sigma$ drop faster. An important
point to pay is that the probability is not conserved. From the corresponding
probability distribution picture in Fig. 8(b), the behavior of $F_{p}$ is the
similar for different effective temperatures. A single peak emerges near $t=0$
in the curve of $F_{p}$ for the given effective temperature. This can be
understood as a large number of first passage events occur in a short interval
of time, and the probability distribution decays exponentially with time. The
effect of temperature at phase transition points on $F_{p}$ is consistent with
that of $\Sigma$ and $G$. That means the higher effective temperature, the
faster probability decreases, the easier phase transition occurs, and the
lower depth of the barrier is; otherwise the lower temperature, the slower
probability decreases, the harder phase transition occurs, and the higher
depth of the barrier is.

##  V discussions and conclusions

In this paper, we mainly have focused on the extended thermodynamic phase
transition of the four-dimensional topological dS spacetime with the non-
linear charge correction, which can be regarded as an ordinary thermodynamic
system in the thermodynamic equilibrium.

Firstly we reviewed the first law of thermodynamics and effective
thermodynamic quantities for the higher-dimensional dS spacetime with the non-
linear charge correction. Note that the volume of this system is the
geometrical volume between the dS black hole horizon and cosmological horizon
and its conjugate thermodynamic quantity is the effective pressure, instead of
the cosmological constant. Then we analyzed the thermodynamic property of
phase transition for the four-dimensional dS spacetime with different non-
linear charge corrections. We found the critical radio, horizon (of the dS
black hole), and effective pressure are all increasing monotonically with the
charge correction $\bar{\phi}$, while the critical radius (for the
cosmological horizon), volume, effective temperature, and entropy are
decreasing monotonically. Especially, with the increasing of the non-linear
charge correction the two horizons get closer and closer, and the correction
entropy $\bar{f}(x)$ is negative and indicates the interaction between the two
horizons stronger and stronger.

In order to obtain the phase transition point, one can use the Maxwell’s
equal-area law or the Gibbs free energy. Here we point out that the results
obtained by these two methods only coincide if the thermodynamic first law
holds. For the low effective temperature, we found there exists the phase
transition in $P_{eff}-V$ with different non-linear charge corrections. From
the view of the Gibbs free energy, we also found the classical swallow tail
behavior in $G_{P_{eff}}-T_{eff}$. The phase transition points in $P_{eff}-V$
and $G_{T_{eff}}-P_{eff}$ are the same for the given effective temperature
$T_{eff}<T_{eff}^{c}$. And there were schottky peaks in
$C_{P_{eff}}-T_{eff},~{}\alpha_{P_{eff}}-x,~{}\kappa_{T_{eff}}-x$ nearby the
critical point. These phase transition properties are similar to that in AdS
black holes. It was unique that the capacity at constant volume is nonzero,
which is completely contrary to that in AdS black holes. We can regard it as a
difference between dS spacetime and AdS black hole.

Then we investigated the dynamic process of phase transition for this system.
From the inspect of Gibbs free energy, we found there is the double-well in
$G-r_{+}$. The two local minimums correspond to the stable large and small dS
black holes and have the same depth. The local maximum stands for the unstable
intermediate dS black hole and acts as a barrier between the stable large and
small dS black holes. It was a signal of emerging phase transition from the
point view of $G$. Next we studied the dynamical process of phase transition
governed by the Forkker-Planck equation. By imposing the reflection boundary
conditions and considering a Gaussian wave packet as the initial condition, we
obtained the numerical result of the Forkker-Planck equation: the initial
Gaussian wave packet at the stable large or small dS black holes decreases
with increasing with time, however at the same time the other peak of
$\rho(t,r_{+})$ at the stable small or large dS black holes increases from
zero to a same constant. That indicates that with increasing time the system
will leave from the initial state to another state, until it becomes a
coexistent state, which is consistent with the fact that the depth of two
wells of $G$ are the same value.

Finally we considered the first passage time. By imposing the absorption
boundary condition on the intermediate-potential black hole state and
considering a Gaussian wave packet at the stable large and small dS black
holes as the initial condition, we also obtained the numerical result of the
Forkker-Planck equation: $\Sigma$ decays very fast in a short time and it
drops faster with increasing effective temperature. And the behavior of
$F_{p}$ is the similar for different effective temperatures. There exists a
single peak near $t=0$ in $F_{p}$. This can be understood as a large number of
first passage events occur in a short interval of time, and the probability
distribution decays exponentially with time. From the effect of effective
temperature on $F_{p}$, $\Sigma$, and $G$, we found the higher temperature,
the faster probability decreases, the easier phase transition occurs, and the
lower depth of the barrier is; otherwise the lower temperature, the slower
probability decreases, the harder phase transition occurs, and the higher
depth of the barrier is.

## Acknowledgments

We would like to thank Prof. Zong-Hong Zhu, Ren Zhao, and Meng-Sen Ma for
their indispensable discussions and comments. This work was supported by the
National Natural Science Foundation of China (Grant No. 11705106, 11475108,
11605107, 12075143), the Natural Science Foundation of Shanxi Province, China
(Grant No.201901D111315), the Natural Science Foundation for Young Scientists
of Shanxi Province, China (Grant No.201901D211441), the Scientific Innovation
Foundation of the Higher Education Institutions of Shanxi Province (Grant Nos.
2020L0471, 2020L0472), and the Science Technology Plan Project of Datong City,
China (Grant Nos. 2020153).

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#  $\lambda$–Core Distance Partitions

Xandru Mifsud  Faculty of Science, University of Malta, Msida, Malta
<EMAIL_ADDRESS>
###### Abstract

The $\lambda$–core vertices of a graph correspond to the non–zero entries of
some eigenvector of $\lambda$ for a universal adjacency matrix $\mathbf{U}$ of
the graph. We define a partition of the vertex set $V$ based on the
$\lambda$–core vertex set and its neighbourhoods at a distance $r$, and give a
number of results relating the structure of the graph to this partition. For
such partitions, we also define an entropic measure for the information
content of a graph, related to every distinct eigenvalue $\lambda$ of
$\mathbf{U}$, and discuss its properties and potential applications.

###### keywords:

Universal Adjacency Matrices, Core Vertices, Graph Labelling , Information
Content

###### MSC:

[2010] 05C50, 15A18

margin=1in

Published in Linear Algebra and Its Applications,
https://doi.org/10.1016/j.laa.2020.12.012

##  1 Introduction

###  1.1 Nomenclature

An undirected graph $G=(V,E)$ has a finite vertex set
$V=\\{v_{1},\dots,v_{n}\\}$ and an edge set $E$ of 2–element subsets of $V$.
We use $n$ exclusively to denote the number of vertices. The graphs we
consider are connected and without loops or multi–edges. We predominantly use
the definitions of Cvetković et al. [2009], with notation adapted accordingly.

The adjacency matrix of a graph $G$ is the $n\times n$ matrix
$\mathbf{A}=(a_{ij})$ such that $a_{ij}=1$ if the vertices $v_{i}$ and $v_{j}$
are adjacent and $a_{ij}=0$ otherwise. The degree matrix is the $n\times n$
matrix $\mathbf{D}=\text{diag}\left(\rho(v_{1}),\dots,\rho(v_{n})\right)$,
where $\rho(v)$ is the number of edges incident to $v$. Let $\mathbf{I}$ and
$\mathbf{J}$ denote the $n\times n$ identity and all-ones matrices,
respectively. Haemers and Omidi [2011] defined the universal adjacency matrix
$\mathbf{U}=\gamma_{A}\mathbf{A}+\gamma_{D}\mathbf{D}+\gamma_{I}\mathbf{I}+\gamma_{J}\mathbf{J}$,
for some scalars $\gamma_{A}\neq 0,\gamma_{D},\gamma_{I}$ and $\gamma_{J}$.
Depending on the choice of parameters, $\mathbf{U}$ encodes a number of
well–known and important matrices in graph theory, most notably the adjacency
and Laplacian matrices.

The eigenvalue spectrum of a symmetric $n\times n$ matrix $\mathbf{M}$ is a
multi–set denoted by ${\rm spec}(\mathbf{M})$, where for
$\lambda^{\left(m_{\mathbf{M}}(\lambda)\right)}\in{\rm spec}(\mathbf{M})$,
$\lambda$ is the eigenvalue and $m_{\mathbf{M}}(\lambda)$ is the multiplicity
of $\lambda$ in the spectrum. For brevity, we shall (with some abuse of
notation) simply write $\lambda\in{\rm spec}(\mathbf{M})$.

A vertex $v_{i}$ of a graph $G$ is a $\lambda$–core vertex for $\mathbf{U}$ if
there exists an eigenvector $\mathbf{x}$ of $\mathbf{U}$ such that
$\mathbf{Ux}=\lambda\mathbf{x}$ and the $i^{\rm th}$ entry of $\mathbf{x}$ is
non–zero. Otherwise, $v_{i}$ is said to be a $\lambda$–core–forbidden vertex.
The sets of $\lambda$–core and $\lambda$–core–forbidden vertices for
$\mathbf{U}$ are denoted by $CV_{\lambda}$ and $CFV_{\lambda}$ respectively.
Diagrammatically, we draw v to represent $\lambda$–core vertices, u for
vertices which have a $\lambda$–core neighbour but are not $\lambda$–core
themselves, and w otherwise.

The terms core and core–forbidden vertices were first introduced by Sciriha
[1997] for the kernel of $\mathbf{A}$. Similar concepts exist: for example, in
linear algebra the core–forbidden vertices that increase the multiplicity of
$\lambda$ are termed as Parter vertices (after the work of S. Parter during
the 1960s), while those that leave the multiplicity unchanged are called
neutral vertices. The core vertices are termed as downer vertices, since they
decrease the multiplicity of $\lambda$. In the context of the Laplacian, the
core–forbidden vertices are typically known as the Fiedler vertices, after the
pioneering work by M. Fiedler throughout the 1970s and 1980s.

More recently, new terminology has been introduced for different contexts. Of
significant importance are the $0$–core and $0$–core–forbidden vertices, which
have been studied greatly for singular trees. The nullity $\eta(G)$ of a graph
is the dimension of the kernel of $\mathbf{A}$. In Fernandes and da Cruz
[2018], the $0$–core–forbidden vertices of a tree are termed as $F$–vertices.
Those vertices that, in particular, increase the nullity of $T$ or the
multiplicity of $0$ for the Laplacian of $T$, are termed as $P$–vertices.

###  1.2 Overview

For a graph $G$ with independent $\lambda$-core vertices, the set of their
neighbours $N(CV_{\lambda})$ is disjoint from $CV_{\lambda}$. Hence one can
define a partition of $V$ with blocks $CV_{\lambda}$, $N(CV_{\lambda})$, and
$CFV_{R,\lambda}$ (which is the set of remaining vertices not in
$CV_{\lambda}$ and $N(CV_{\lambda})$). Such a partitioning was studied closely
for the eigenvalue $\lambda=0$ of $\mathbf{A}$ by Sciriha et al. [2020] for
graphs with independent $CV_{0}$, and by Jaume and Molina [2018] for the case
of singular trees (which always have an independent $CV_{0}$). In Section 2,
we introduce the $\lambda$–Core Distance Partition ($\lambda$–CDP), which is a
natural generalisation for the case when $CV_{\lambda}$ is not an independent
set and hence $N(CV_{\lambda})$ is not disjoint from $CV_{\lambda}$. Moreover,
we study the relation between the $\lambda$–CDP and the symmetries of the
graph, and how the core distance partitions of two distinct eigenvalues relate
to each other.

In Section 3 we define an index for the information content of a graph related
to its $\lambda$–CDP, and study its lower and upper bounds for a fixed number
of vertices $n$ and varying number of $\lambda$–core vertices,
$|CV_{\lambda}|$. Lastly in Section 4 we consider a number of applications
related to this index and the structure of graphs with a singular adjacency
matrix, most notably molecular graphs.

##  2 $\lambda$–Core Distance Partitions

Consider a connected graph $G$, and
$\lambda\in\text{spec}\left(\mathbf{U}\right)$. The function $d(u,v)\colon
V\times V\rightarrow\mathbb{N}\ \dot{\cup}\ \\{0\\}$ gives the length of the
shortest path between any two vertices $u$ and $v$. We define the minimum
distance from any vertex $v\in V$ to some $\lambda$–core vertex $u\in
CV_{\lambda}$ by the function $d_{\lambda}(v)\colon V\rightarrow\mathbb{N}\
\dot{\cup}\ \\{0\\}$,

$d_{\lambda}(v)=\min\left\\{d(v,u)\colon u\in CV_{\lambda}\right\\}.$ (2.1)

The non–negative integer $D_{\lambda}=\max\left\\{d_{\lambda}(v)\colon v\in
V\right\\}$ is the maximum distance assigned by $d_{\lambda}$ to a vertex,
which is bounded above by the diameter of the graph. We define the set of
vertices at a minimum distance $0\leq i\leq D_{\lambda}$ from $\lambda$–core
vertex as the set $V_{\lambda,i}=\left\\{v\in V\colon
d_{\lambda}(v)=i\right\\}$. It follows then that $CV_{\lambda}=V_{\lambda,0}$.

######  Proposition 2.1 ($\lambda$–Core Distance Partition).

Let $\lambda\in{\rm spec}(\mathbf{U})$. The collection
$\mathcal{V}_{\lambda}=\left\\{V_{\lambda,0},\dots,V_{\lambda,D_{\lambda}}\right\\}$
is a partition of $V$.

###### Proof.

Since $G$ is connected, there exists a path between any pair of vertices. In
particular, every vertex $v\in V$ has a shortest path to some $\lambda$–core
vertex. Hence $d_{\lambda}$ exists for every vertex $v\in V$ and therefore $v$
belongs to some $V_{\lambda,i}\in\mathcal{V}_{\lambda}$, for $0\leq i\leq
D_{\lambda}$. Clearly then, $V$ is the union of
$V_{\lambda,0},\dots,V_{\lambda,D_{\lambda}}$.

Moreover, since $d_{\lambda}$ is clearly not one–to–many, we have that the
intersection of $V_{\lambda,i}$ and $V_{\lambda,j}$ is empty, for $i\neq j$.
The result follows. ∎

######  Remark 2.2.

Note that $d_{\lambda}$ gives an equivalence relation, where two vertices $u$
and $v$ are said to be related if $d_{\lambda}(u)=d_{\lambda}(v)$. Hence the
$\lambda$–CDP $\mathcal{V}_{\lambda}$ is in fact a collection of equivalence
classes.

###  2.1 $\lambda$–Core Distance Partitions and Orbit Structure

The automorphism group $\text{Aut}(G)$ of a graph is the group from $G$ onto
itself. If $v_{i}$ is adjacent to $v_{j}$ then for any automorphism
$\sigma\in\text{Aut}(G)$, $\sigma(v_{i})$ is adjacent to $\sigma(v_{j})$. The
orbit of $\text{Aut}(G)$ on $v\in V$ is the set
$\pi(v)=\left\\{\sigma(v)\colon\sigma\in\text{Aut}(G)\right\\}$. We denote the
set of orbits by $\Pi$. A graph $G$ is vertex–transitive if it has a single
orbit. The orbits of $\text{Aut}(G)$ induce a partition
$\Pi=\left\\{\pi_{1},\pi_{2},\dots,\pi_{s}\right\\}$ of $V$, known as the
orbital partition of $G$.

123456789101112$V_{1,0}$$V_{1,1}$$V_{1,2}$G Figure 1: $1$–CDP of the graph
$G$, for $\mathbf{U}=\mathbf{A}$.

Consider the cubic graph $G$ in Figure 1 as a motivating example. The spectrum
of its adjacency matrix (up to 3 decimal places) is:
$\left\\{3^{(1)},2.115^{(1)},2^{(1)},1.303^{(1)},1^{(1)},-0.254^{(1)},-1^{(3)},-1.861^{(1)},-2^{(1)},-2.303^{(1)}\right\\}.$
For $\lambda=1$, the $1$–CDP for $\mathbf{U}=\mathbf{A}$ is:

$\mathcal{V}_{1}\ =\left\\{V_{1,0}=\\{1,2,3,4,5,6\\},\ V_{1,1}=\\{7,8,9\\},\
V_{1,2}=\\{10,11,12\\}\right\\}$

Moreover, the orbits of ${\rm Aut}(G)$ acting on $V$ are,

$\Pi=\left\\{\pi_{1}=\\{1,3,4,6\\},\pi_{2}=\\{2,5\\},\pi_{3}=\\{7,9\\},\pi_{4}=\\{8\\},\pi_{5}=\\{10,12\\},\pi_{6}=\\{11\\}\right\\}$

Notice that each block in the $1$–CDP is a disjoint union of orbits in $\Pi$.
In this section we will show that this holds for every $\lambda$–CDP of any
universal adjacency matrix $\mathbf{U}$ of $G$.

Given a permutation $\sigma\in\text{Aut}(G)$, there exists a corresponding
permutation matrix $\mathbf{P_{\sigma}}$ such that
$\mathbf{P_{\sigma}A}=\mathbf{AP_{\sigma}}$. Since $\mathbf{D}$ and
$\mathbf{I}$ have constant entries with respect to each orbit, and since
$\mathbf{J}$ is a constant matrix, then $\mathbf{P}_{\sigma}$ also commutes
with $\mathbf{D}$, $\mathbf{I}$, and $\mathbf{J}$. Since matrix multiplication
is distributive, then $\mathbf{U}$ and $\mathbf{P}_{\sigma}$ commute:
$\mathbf{U}\mathbf{P}_{\sigma}=\mathbf{P}_{\sigma}\mathbf{U}$. The following
result is due to Akbari et al. [2013].

######  Proposition 2.3 (Akbari et al. [2013]).

Let $G$ be vertex–transitive and $\lambda\in{\rm spec}(\mathbf{A})$. Then
every vertex of $G$ is a $\lambda$–core vertex.

The following proposition is a generalisation of Proposition 2.3, of which it
is a special case for when $G$ is vertex–transitive and hence has a single
orbit.

######  Proposition 2.4.

Let $\lambda\in{\rm spec}(\mathbf{U})$ and let $u,v$ belong to the same orbit
$\pi\in\Pi$. If $u$ is a $\lambda$–core vertex, then $v$ is also a
$\lambda$–core vertex.

###### Proof.

Since $u$ and $v$ belong to the same orbit, it follows by the definition of an
orbit that there exists a permutation $\sigma\in\text{Aut}(G)$, such that
$\sigma(u)=v$. Let $\mathbf{P_{\sigma}}$ be the corresponding permutation
matrix. Since $u$ is $\lambda$–core, there exists an eigenvector $\mathbf{x}$
for $\lambda$ such that the $u^{\rm th}$ entry of $\mathbf{x}$ is non–zero.
Now,
$\mathbf{UP_{\sigma}x}=\mathbf{P_{\sigma}Ux}=\lambda\mathbf{P_{\sigma}x}$,
since $\mathbf{UP_{\sigma}}=\mathbf{P_{\sigma}U}$ and
$\mathbf{Ux}=\lambda\mathbf{x}$. Hence $\mathbf{P_{\sigma}x}$ is an
eigenvector of $\mathbf{U}$ as well. Since $\mathbf{P_{\sigma}}$ permutes the
$u^{\text{th}}$ entry of $\mathbf{x}$ with the $v^{\text{th}}$ entry, it
follows that the $v^{\rm th}$ entry of $\mathbf{P_{\sigma}x}$ is non–zero.
Therefore $v$ is a $\lambda$–core vertex. ∎

######  Remark 2.5.

It follows that all the vertices in the same orbit are either $\lambda$–core
or $\lambda$–core forbidden. Hence, $CV_{\lambda}$ is the disjoint union of
some collection of orbits
$\left\\{\pi_{i_{1}},\dots,\pi_{i_{s}}\right\\}\subseteq\Pi$.

######  Theorem 2.6.

Let $\lambda\in{\rm spec}(\mathbf{U})$ and $0\leq i\leq D_{\lambda}$. Then
$V_{\lambda,i}$ is the disjoint union of a collection of orbits
$\left\\{\pi_{1},\dots,\pi_{s_{i}}\right\\}\subseteq\Pi$.

###### Proof.

That $V_{\lambda,0}=CV_{\lambda}$ is the disjoint union of orbits is discussed
in Remark 2.5. Suppose that up to some $0\leq i<D_{\lambda}$, $\forall 0\leq
k\leq i,V_{\lambda,k}$ is the disjoint union of some collection of orbits
$\Pi_{k}=\left\\{\pi^{(k)}_{1},\dots,\pi^{(k)}_{s_{k}}\right\\}\subseteq\Pi$.
We show that the claim holds for $i+1$.

Suppose that $V_{\lambda,i+1}$ is not the disjoint union of some collection of
orbits. Then there exists two vertices $u$ and $v$ in an orbit $\pi$, such
that $v$ is in $V_{\lambda,i+1}$ and $u$ is in $V_{\lambda,j}$, for some
$i+2\leq j\leq D_{\lambda}$. Note that $\pi$ is distinct from those in the
collection $\dot{\cup}_{k=0}^{i}\Pi_{k}$.

By definition of the $\lambda$–CDP, $v$ in $V_{\lambda,i+1}$ must have a
neighbour in $V_{\lambda,i}$. Moreover, there must exist a permutation
$\sigma\in{\rm Aut}(G)$ which maps $v$ to $u$ while preserving adjacency. In
particular, $V_{\lambda,k}$ for $0\leq k\leq i$ are preserved since they are
the disjoint union of orbits, and hence $u$ must have a neighbour in
$V_{\lambda,i}$. Since $i+1<j$, then $u$ has a path to some $\lambda$–core
vertex of length less than $j$, and hence cannot be in $V_{\lambda,j}$, a
contradiction. Therefore $V_{\lambda,i+1}$ is the disjoint union of some
collection of orbits, completing the proof. ∎

###  2.2 Relating the Core Distance Partitions of Two Distinct Eigenvalues

We continue our study of the relation between the spectrum of $\mathbf{U}$ and
the structure of $G$, in light of the $\lambda$–CDP. We say that for two
distinct eigenvalues $\lambda$ and $\mu$, the $\mu$–CDP can be constructed
from the $\lambda$–CDP if every block in the $\mu$–CDP is a disjoint union of
blocks in the $\lambda$–CDP. In particular, we are interested in the following
problem.

######  Problem 2.7.

Let $\lambda\in{\rm spec}(\mathbf{U})$ and let the $\lambda$–CDP be given.
When does there exist an eigenvalue $\mu\neq\lambda$ of $\mathbf{U}$, such
that the $\mu$–CDP can be constructed from the $\lambda$–CDP?

As it turns out, if the $\lambda$–CDP is an equitable partition, then under
certain circumstances reconstruction is possible. First however we require
some preliminary definitions and results. The characteristic matrix
$\mathbf{C}$ of a partition $\\{V_{1},\dots,V_{k}\\}$ is an $n\times k$
matrix, where the $i^{\rm th}$ column $\mathbf{c}_{i}$ is associated with the
block $V_{i}$, such that for every vertex in $V_{i}$ the corresponding entry
in $\mathbf{c}_{i}$ is 1, and 0 otherwise. An equitable partition with $k$
blocks is a partition of $V$ such that there exists a $k\times k$ matrix
$\mathbf{B}$, known as the divisor matrix, satisfying
$\mathbf{U}\mathbf{C}=\mathbf{C}\mathbf{B}$.

######  Remark 2.8.

Let $\mathbf{x}$ be an eigenvector for $\mu$ of $\mathbf{B}$ such that its
$i^{\rm th}$ entry $x_{i}$ is non–zero. If $\mathbf{C}\mathbf{x}$ is also an
eigenvector for some $\lambda\in{\rm spec}(\mathbf{U})$, then all the vertices
in $V_{i}$ must be $\lambda$–core vertices, since the corresponding entry in
$\mathbf{C}\mathbf{x}$ is $x_{i}$ by definition of $\mathbf{C}$.

It is well–known that for the case when $\mathbf{U}=\mathbf{A}$, the
characteristic polynomial of $\mathbf{B}$ divides the characteristic
polynomial of $\mathbf{A}$ (Theorem 3.9.5, Cvetković et al. [2009]). More
generally, the result holds for any universal adjacency matrix $\mathbf{U}$,
which we state as follows without proof.

######  Theorem 2.9.

Let $\mathcal{C}$ be an equitable partition of $V$ with divisor and
characteristic matrices $\mathbf{B}$ and $\mathbf{C}$, respectively. The
characteristic polynomial of $\mathbf{B}$ divides the characteristic
polynomial of $\mathbf{U}$.

######  Remark 2.10.

Consider some $\lambda$–CDP $\mathcal{V}_{\lambda}$ to be equitable. By
combining Remark 2.8 and Theorem 2.9 we observe that for every eigenvalue of
$\mathbf{U}$ that is also an eigenvalue of $\mathbf{B}$, the core vertex set
has a subset which is the disjoint union of some blocks in
$\mathcal{V}_{\lambda}$.

######  Theorem 2.11.

Let $\lambda\in{\rm spec}(\mathbf{U})$ and $\mathcal{V}_{\lambda}$ be an
equitable partition with divisor and characteristic matrices $\mathbf{B}$ and
$\mathbf{C}$, respectively. Let $\mu\in{\rm spec}(\mathbf{B})$. Then there
exists a subset of $CV_{\mu}$ which is the disjoint union of a finite
collection of blocks in $\mathcal{V}_{\lambda}$. Moreover, if
$m_{\mathbf{U}}(\mu)=m_{\mathbf{B}}(\mu)$ then equality holds between the two
sets and $\mathcal{V}_{\mu}$ can be constructed from $\mathcal{V}_{\lambda}$.

###### Proof.

Let $B$ be a basis for the eigenspace of $\mu$ for $\mathbf{B}$. Define the
indexing set $X_{B}$ of row–indices corresponding to some non–zero row of any
vector $\mathbf{b}$ in $B$:
$X_{B}=\left\\{i\in\left\\{1,\dots,D_{\lambda}+1\right\\}:\exists\mathbf{b}\in
B,\mathbf{b}\cdot\mathbf{e}_{i}\neq 0\right\\}$. By definition of an equitable
partition, each index $i$ in $X_{B}$ corresponds to the block
$V_{\lambda,i-1}$ of $\mathcal{V}_{\lambda}$. As discussed in Remark 2.10 we
have that $\underset{i\in X_{B}}{{\dot{\cup}}}V_{\lambda,i-1}\subseteq
CV_{\mu}$ and the first part of the result follows.

Let the basis $B$ be $\\{\mathbf{b}_{1},\dots,\mathbf{b}_{r}\\}$ and consider
the set
$B^{\prime}=\\{\mathbf{C}\mathbf{b}_{1},\dots,\mathbf{C}\mathbf{b}_{r}\\}$.
Since $B$ is an independent set, then $B^{\prime}$ must also be an independent
set. By Remark 2.8 and Theorem 2.9, $B^{\prime}$ is a basis for some sub–space
of the eigenspace of $\mu$ for $\mathbf{U}$. In the case that
$m_{\mathbf{U}}(\mu)=m_{\mathbf{B}}(\mu)$, it follows that $B^{\prime}$ is a
basis for the eigenspace of $\mu$ for $\mathbf{U}$. Since every vertex in
$CV_{\mu}$ must correspond to a non–zero entry for some eigenvector in
$B^{\prime}$, it follows that $\underset{i\in
X_{B}}{{\dot{\cup}}}V_{\lambda,i-1}=CV_{\mu}$. Let $X_{B}=X^{0}_{B}$. For
$d\in\mathbb{N}$, recursively define,

$X^{d}_{B}=\left\\{i\in\\{1,\dots,D_{\lambda}+1\\}:\exists j\in
X^{d-1}_{B},i=j\pm 1\right\\}\backslash\bigcup\limits_{k=0}^{d-1}X^{k}_{B}$

which is the set of indices, not already appearing in some $X^{j}_{B}$ for
$0\leq j<d$, of blocks whose vertices are at a distance 1 from the vertices in
the blocks with indices in $X^{d-1}_{B}$. Hence the union of the blocks
$V_{\lambda,i-1}$ with indices in $X^{d}_{B}$ gives the vertices at a distance
$d$ from $CV_{\mu}$. Moreover, $\exists D_{\mu}\in\mathbb{N}$ such that
$X_{B}^{D_{\mu}}\neq\emptyset$ and for every $d>D_{\mu}$,
$X_{B}^{d}=\emptyset$. Hence $\mathcal{V}_{\mu}$ can be constructed from the
indexing sets $X^{0}_{B},\dots,X^{D_{\mu}}_{B}$. ∎

####  2.2.1 Worked Example

Consider once again the cubic graph $G$ in Figure 1. The $1$–CDP is also an
equitable partition, having the following divisor and characteristic matrices
$\mathbf{B}$ and $\mathbf{C}$, respectively,

$\displaystyle\mathbf{B}=\left(\begin{array}[]{ccc}2&1&0\\\ 2&0&1\\\
0&1&2\end{array}\right)$
$\displaystyle\mathbf{C}=\left(\begin{array}[]{cccccccccccc}1&1&1&1&1&1&0&0&0&0&0&0\\\
0&0&0&0&0&0&1&1&1&0&0&0\\\ 0&0&0&0&0&0&0&0&0&1&1&1\\\
\end{array}\right)^{\intercal}.$

The spectrum of $\mathbf{B}$ is $\\{3^{(1)},2^{(1)},-1^{(1)}\\}$. Consider
$\mu=2$, which is in both ${\rm spec}(\mathbf{A})$ and ${\rm
spec}(\mathbf{B})$, with the same multiplicity (equal to 1). The basis of the
eigenspace for $2$ of $\mathbf{B}$ is $\left\\{(-1,0,2)^{\intercal}\right\\}$,
which is non–zero with respect to each block of $\mathcal{V}_{1}$. On the
other hand, the eigenspace for $2$ of $\mathbf{A}$ is spanned by a single
vector, $\left(-1,-1,-1,-1,-1,-1,0,0,0,2,2,2\right)^{\intercal}$. In this
case, $CV_{2}$ is the disjoint union of the blocks $V_{1,0}$ and $V_{1,2}$
from $\mathcal{V}_{1}$. Moreover, $\mathcal{V}_{2}$ can be constructed from
$\mathcal{V}_{1}$: $\mathcal{V}_{2}\ =\left\\{V_{2,0}=V_{1,0}\ \dot{\cup}\
V_{1,2},\ V_{2,1}=V_{1,1}\right\\}$.

##  3 Information Content of Graphs and Topological Indices

###  3.1 Information Content and Structure of $\lambda$–Core Distance
Partitions

The study of the information content of graphs is an important and active
field, which is at the intersection of graph and information theory. The
notion of information content of graphs was first introduced by Rashevsky
[1955], Trucco [1956], and was generalised and formalised by Mowshowitz [1968]
as follows: Given a graph $G$ with a vertex set $V$ and a partition of $V$
into equivalence classes, a finite probability scheme can be assigned to this
partitioning, by assigning the probability $\frac{\left|V_{i}\right|}{n}$ to
each equivalence class $V_{i}$. An entropic measure associated with this
finite probability scheme could then be defined as
$I=-\sum_{i=0}^{k}\frac{\left|V_{i}\right|}{n}\log\left(\frac{\left|V_{i}\right|}{n}\right)$,
which is based on Shannon’s Entropy. Such measures form part of a broader
class of indices, known as topological indices. Note that the logarithms we
consider are typically of base 2, $e$ or 10, all of which are monotonically
increasing.

Since the orbital partition $\Pi$ is in fact a collection of equivalence
classes, Mowshowitz studied the entropy measure $I_{a}$ associated with $\Pi$:
$I_{a}=-\sum_{i=0}^{s}\frac{\left|\pi_{i}\right|}{n}\log\left(\frac{\left|\pi_{i}\right|}{n}\right)$.
Similarly, for $\lambda\in{\rm spec}(\mathbf{U})$, the $\lambda$–CDP
$\mathcal{V}_{\lambda}$ is a collection of equivalence classes, as discussed
in Remark 2.2, and hence a finite probability scheme may be associated with
it. In this manner, we can define a topological index for every distinct
eigenvalue of $\mathbf{U}$, related to the structure of the neighbourhoods
$V_{\lambda,i}$ of $CV_{\lambda}$ at a distance $i$, and their size.

######  Theorem 3.1.

Let $\lambda\in{\rm spec}(\mathbf{U})$ and $\mathcal{V}_{\lambda}$ be the
associated $\lambda$–CDP. Then for each equivalence class $V_{\lambda,i}$ in
$\mathcal{V}_{\lambda}$, we can associate the probability
$\frac{\left|V_{\lambda,i}\right|}{n}$, defining a finite probability scheme
with entropy

$I_{\lambda}=-\sum\limits_{i=0}^{D_{\lambda}}\dfrac{\left|V_{\lambda,i}\right|}{n}\log\left(\dfrac{\left|V_{\lambda,i}\right|}{n}\right).$

###  3.2 Relating $I_{\lambda}$ and $I_{\mu}$ for Two Distinct Eigenvalues

Consider a value $x_{i}$ associated with each block $V_{\lambda,i}$ in some
$\lambda$–CDP. Then,
$W_{\lambda}(x_{0},\dots,x_{D_{\lambda}})=\frac{1}{n}\sum_{i=0}^{D_{\lambda}}\left|V_{\lambda,i}\right|x_{i}$
is the arithmetic mean of these values, weighted by the size of each block.
For brevity, we write $W_{\lambda}\left(\\{x_{i}\\}\right)$, where $0\leq
i\leq D_{\lambda}$. Let $\mathcal{P}$ be a partition of $V$ such that each
block $V_{\lambda,i}$ in $\mathcal{V}_{\lambda}$ is the disjoint union of
$k_{i}\in\mathbb{N}$ blocks in $\mathcal{P}$.

It can be shown that
$-\sum_{i=1}^{|\mathcal{P}|}\frac{|P_{i}|}{n}\log\left(\frac{|P_{i}|}{n}\right)$
is bounded above by $I_{\lambda}+W_{\lambda}\left(\\{\log(k_{i})\\}\right)$.
We first require the following well–known inequality in information theory.

######  Theorem 3.2 (Theorem 2.7.1, Cover and Thomas [2005]).

For non–negative numbers $x_{1},\dots,x_{n}$ and $y_{1},\dots,y_{n}$,

$\sum_{k=1}^{n}x_{k}\log\left(\dfrac{x_{k}}{y_{k}}\right)\geq\left(\sum_{k=1}^{n}x_{k}\right)\log\left(\dfrac{\sum_{k=1}^{n}x_{k}}{\sum_{k=1}^{n}y_{k}}\right)$

with equality if and only if $\frac{x_{k}}{y_{k}}={\rm constant}$.

######  Proposition 3.3.

Let $\lambda\in{\rm spec}(\mathbf{U})$ and $\mathcal{V}_{\lambda}$ be the
associated $\lambda$–CDP. Let $\mathcal{P}$ be a partition of $V$ such that
each block $V_{\lambda,i}$ of $\mathcal{V}_{\lambda}$ is the disjoint union of
$k_{i}\in\mathbb{N}$ blocks of $\mathcal{P}$. Then,

$-\sum_{i=1}^{|\mathcal{P}|}\frac{|P_{i}|}{n}\log\left(\frac{|P_{i}|}{n}\right)\leq
I_{\lambda}+W_{\lambda}\left(\\{\log(k_{i})\\}\right)$ (3.1)

###### Proof.

Let $V_{\lambda,i}$ be the disjoint union of $k_{i}\in\mathbb{N}$ blocks
$\left\\{P^{(i)}_{1},\dots,P^{(i)}_{k_{i}}\right\\}$ in $\mathcal{P}$. Then,

$\displaystyle\phantom{{}=1}\dfrac{1}{n}\sum_{k=1}^{k_{i}}\left|P_{k}^{(i)}\right|\log\left(\tfrac{\left|P_{k}^{(i)}\right|}{n}\right)$
$\displaystyle\text{since}\ V_{\lambda,i}\ \text{is the union
of}\left\\{P^{(i)}_{1},\dots,P^{(i)}_{k_{i}}\right\\},$
$\displaystyle\geq\dfrac{1}{n}\left(\sum_{k=1}^{k_{i}}\left|P_{k}^{(i)}\right|\right)\log\left(\tfrac{\sum_{k=1}^{k_{i}}\left|P_{k}^{(i)}\right|}{nk_{i}}\right)$
$\displaystyle\text{by Theorem \ref{logsum} for}\
x_{k}=\left|P_{k}^{(i)}\right|,y_{k}=n,$
$\displaystyle=\tfrac{|V_{\lambda,i}|}{n}\left[\log\left(\tfrac{|V_{\lambda,i}|}{n}\right)-\log(k_{i})\right]$
$\displaystyle\text{by properties of logs and}\
|V_{\lambda,i}|=\\!\sum_{k=1}^{k_{i}}\left|P_{k}^{(i)}\right|.$

Applying this inequality to each term in $I_{\lambda}$, the bound follows
immediately. ∎

Such an inequality is closely associated with Problem 2.7. For two distinct
eigenvalues $\lambda,\mu\in{\rm spec}(\mathbf{U})$, such that the $\mu$–CDP
can be constructed from the $\lambda$–CDP, the following result follows
immediately.

######  Proposition 3.4.

Let $\lambda,\mu\in{\rm spec}(\mathbf{U})$ be two distinct eigenvalues. If the
$\mu$–CDP can be constructed from the $\lambda$–CDP, then there exist
$|D_{\mu}|+1$ positive integers $k_{i}$ such that $I_{\lambda}\leq
I_{\mu}+W_{\mu}\left(\\{\log(k_{i})\\}\right)$.

In particular, as an immediate consequence of Theorem 2.11, we have the
following corollary.

######  Corollary 3.5.

Let $\lambda\in{\rm spec}(\mathbf{U})$ and $\mathcal{V}_{\lambda}$ be an
equitable partition with divisor and characteristic matrices $\mathbf{B}$ and
$\mathbf{C}$, respectively. Let $\mu\in{\rm spec}(\mathbf{B})$ such that
$m_{\mathbf{U}}(\mu)=m_{\mathbf{B}}(\mu)$. Then $I_{\lambda}\leq
I_{\mu}+W_{\mu}\left(\\{\log(k_{i})\\}\right)$.

Consider the $1$–CDP and $2$–CDP for the adjacency matrix $\mathbf{A}$ of the
graph $G$ in Figure 1. As in the worked example of Section 2.2.1, the $2$–CDP
can be constructed from the $1$–CDP. In particular, the $2$–CDP has two blocks
$V_{2,0}$ and $V_{2,1}$, which are the disjoint union of $k_{0}=2$ and
$k_{1}=1$ blocks from the $1$–CDP, respectively. Up to 4 decimal places, we
have that $W_{2}\left(\ln(2),\ln(1)\right)\simeq 0.5199$ and $I_{2}\simeq
0.5623$. Moreover, $I_{1}\simeq 1.0397$ and hence $I_{1}\leq
I_{2}+W_{2}\left(\ln(2),\ln(1)\right)\simeq 1.0822$.

###  3.3 Note on the Minimum–Maximum Attainable Bounds of $I_{\lambda}$

Of interest are the minimum and maximum bounds attainable by $I_{\lambda}$ for
a fixed number of vertices $n$, and varying number of $\lambda$–core vertices.
These bounds are given in the following theorem, and visualised in Figure 2.

######  Proposition 3.6.

Let $\lambda\in{\rm spec}(\mathbf{U})$ and let $|CV_{\lambda}|=k,1\leq k\leq
n$. Then,

${\left.\begin{array}[]{cr}0&k=n\\\
-\tfrac{n-k}{n}\log\left(\tfrac{n-k}{n}\right)&k<n\end{array}\right\\}}\leq
I_{\lambda}+\dfrac{k}{n}\log\left(\dfrac{k}{n}\right)\leq-\dfrac{n-k}{n}\log\left(\dfrac{1}{n}\right)$

###### Proof.

Let $|CV_{\lambda}|=k$, where $1\leq k<n$. Then $1\leq D_{\lambda}\leq n-k$.
Consider,

$I_{\lambda}+\tfrac{k}{n}\log\left(\tfrac{k}{n}\right)=-\sum_{i=1}^{D_{\lambda}}\sum_{j=1}^{|V_{\lambda,i}|}\tfrac{1}{n}\log\left(\tfrac{|V_{\lambda,i}|}{n}\right)\leq-\sum_{i=1}^{D_{\lambda}}\sum_{j=1}^{|V_{\lambda,i}|}\tfrac{1}{n}\log\left(\tfrac{1}{n}\right)$

since $-\log(x)$ is monotonically decreasing and $1\leq|V_{\lambda,i}|$. Since
$n-k=\sum_{i=1}^{D_{\lambda}}|V_{\lambda,i}|$, the upper–bound
$I_{\lambda}+\tfrac{k}{n}\log\left(\tfrac{k}{n}\right)\leq\tfrac{n-k}{n}\log\left(\tfrac{1}{n}\right)$
follows. For $1\leq k<n$, by a similar argument we have
$-\tfrac{n-k}{n}\log\left(\tfrac{n-k}{n}\right)\leq
I_{\lambda}+\tfrac{k}{n}\log\left(\tfrac{k}{n}\right)$. For the case that
$k=n$, then $I_{\lambda}=\tfrac{n}{n}\log\left(\tfrac{n}{n}\right)=0$,
completing the proof. ∎

$1$$5$$\frac{n}{2}$$n-4$$n$$|CV_{\lambda}|$$I_{\lambda}$$\max\
I_{\lambda}$$\min\ I_{\lambda}$ Figure 2: Discrete plot of the range of
$I_{\lambda}$ for fixed $n$ and varying $|CV_{\lambda}|$, between the
Minimum–Maximum bounds in Proposition 3.6. For $n$ even (as shown here), the
maximum lower–bound occurs at $n/2$.

######  Remark 3.7.

There are a number of observations that one can make on the range of values
that $I_{\lambda}$ may take as the number of $\lambda$–core vertices changes
and $n$ remains fixed. Firstly, the lower–bound is attained when the number of
blocks in $\mathcal{V}_{\lambda}$ is exactly two, while the upper–bound is
attained when the number of blocks is $n-|CV_{\lambda}|+1$. Hence for fixed
$n$ and $|CV_{\lambda}|$, the entropy $I_{\lambda}$ increases as $D_{\lambda}$
increases. Secondly, the maximum lower–bound is attained when
$\mathcal{V}_{\lambda}$ has two blocks of more or less equal size. Entropy
values below this threshold indicate graphs with either very few or almost all
vertices being $\lambda$–core. Moreover, $D_{\lambda}$ must be very small and
hence few vertices must be distant from a $\lambda$–core vertex.

##  4 Applications

For every eigenvalue $\lambda$ of a universal adjacency matrix $\mathbf{U}$
associated with a graph $G$, a notion of information content related to that
eigenvalue may be defined, based on the structure of the $\lambda$–CDP. We say
that a graph $G$ is singular if its adjacency matrix is singular. Partitions
of the vertex set based on neighbourhoods of $\lambda$–core vertices have been
studied greatly in Sciriha et al. [2020], Jaume and Molina [2018] for singular
graphs with independent $CV_{\lambda}$. Such partitions, as well as
topological indices, have many applications in the study of networks and
molecular graphs.

###  4.1 Singular Graphs

In the study of singular graphs, minimal configurations (MCs) are of
importance, as they are considered to be the building blocks for such graphs,
through graph operations such as coalescence.

######  Definition 4.1 (Sciriha [1998]).

A minimal configuration is a singular graph which is either $K_{1}$ or if
$n\geq 3$ then it has a core–graph $F$ induced by $CV_{0}$ and a periphery
$\mathcal{P}\backslash CV_{0}$ satisfying the following conditions:

  1. (i)

$\eta(G)=1$,

  2. (ii)

$\mathcal{P}=\emptyset$ or the graph induced by $\mathcal{P}$ consists of
isolated vertices,

  3. (iii)

$|\mathcal{P}|+1=\eta(F)$.

In particular, MCs are connected (Theorem 2, Sciriha [1998]). Of interest are
bipartite MCs, which are closely associated with slim graphs. Slim graphs were
defined for the case of independent $0$–core vertices in Definition 8 of
Sciriha et al. [2020]. The following definition generalises the notion of slim
graphs to include those graphs for which $CV_{0}$ is not an independent set.

######  Definition 4.2.

A connected singular graph $G$ with $0$–core distance partition
$\mathcal{V}_{0}$ for $\mathbf{U}=\mathbf{A}$, is a slim graph if $CFV_{0}$ is
precisely $V_{0,1}$.

In the case that $CV_{0}$ is independent, then $CFV_{0}$ is exactly
$N(CV_{0})$ for a slim graph. The following proposition characterises such
graphs as being those graphs for which the entropic measure $I_{0}$ is at a
minimum, for a given number of vertices $n$ and $\lambda$–core vertices
$|CV_{0}|$.

######  Proposition 4.3.

Let $G$ be a connected singular graph. Then, $G$ is a slim graph if and only
if $I_{0}$ is equal to the lower–bound for the given $n$ and $|CV_{0}|$.

###### Proof.

If $G$ is a slim graph, then by definition, for $\mathbf{U}=\mathbf{A}$, we
have that $\mathcal{V}_{0}=\\{V_{0,0}=CV_{0},V_{0,1}\\}$, where
$|V_{0,1}|=n-|CV_{0}|$. Therefore by Proposition 3.6, $I_{0}$ is equal to the
lower–bound for the given $n$ and $|CV_{0}|$. Conversely, let $I_{0}$ be equal
to the lower–bound, given $n$ and $|CV_{0}|$. Suppose that the $0$–CDP for
$\mathbf{U}=\mathbf{A}$ has at least three blocks. Then $I_{0}$ has the term
$-\frac{|CV_{0}|}{n}\log\left(\frac{|CV_{0}|}{n}\right)$, and all the other
terms are of the form $-\frac{n-k}{n}\log\left(\frac{n-k}{n}\right)$, where
$|CV_{0}|<k<n$. Since $-\log(x)$ is strictly monotonically decreasing, then
$-\log\left(\frac{n-|CV_{0}|}{n}\right)<-\log\left(\frac{n-k}{n}\right)$ and
hence $I_{0}$ is greater than the lower–bound, a contradiction. The result
follows. ∎

It was shown by Sciriha et al. [2020] that bipartite MCs, which have an
independent $0$–core vertex set, are in fact slim graphs with partite sets
$V_{1}=CV_{0}$ and $V_{2}=N(CV_{0})$, such that $|CV_{0}|=|N(CV_{0})|+1$. The
following is an immediate consequence of Proposition 4.3.

######  Proposition 4.4.

If $G$ is a bipartite minimal configuration then $I_{0}$ is equal to the
lower–bound for the given $n$ and $|CV_{0}|$.

###  4.2 Chemical Considerations

Let $G$ be the $n$–atomic molecular graph of a conjugated hydrocarbon system.
In Hückel Molecular Orbital Theory, the Hamiltonian for such a system is given
by $\mathbf{H}(\alpha,\beta)=\alpha\mathbf{I}+\beta\mathbf{A}$, where
$\mathbf{A}$ is the adjacency matrix of $G$ and $\alpha,\beta$ are two
parameters. Notice that $\mathbf{H}$ may be represented by a universal
adjacency matrix, by setting $\gamma_{A}=\beta$, $\gamma_{I}=\alpha$, and
$\gamma_{D}=0=\gamma_{J}$. Clearly, $\mathbf{H}(\alpha,\beta)$ and
$\mathbf{A}$ commute and hence have the same eigenvectors. If $\lambda_{i}$ is
an eigenvalue in the ordered spectrum of $\mathbf{A}$, then
$E_{i}=\alpha+\beta\lambda_{i}$ is the corresponding eigenvalue of
$\mathbf{H}(\alpha,\beta)$.

Within the Hückel framework, $E_{i}$ corresponds to the $i^{\rm th}$ energy
level. Each eigenvector corresponds to one of the $n$–molecular orbitals,
where the eigenvector entries give a linear combination of the $n$–atomic
orbitals. The normalised values of these entries represent the probability of
electron occupation at each atomic orbital, for the $i^{\rm th}$ molecular
orbital. For an in–depth treatment of the Hückel framework, see Gutman and
Polansky [1986].

Since $\mathbf{H}(\alpha,\beta)$ takes the form of a universal adjacency
matrix, then the $E_{i}$-CDP is defined for every $E_{i}\in{\rm
spec}(\mathbf{H})$. Hence for the $i^{\rm th}$ energy level, we can associate
a topological index $I_{E_{i}}$. Such an index $I_{E_{i}}$ has appealing
interpretations, related to the probabilistic distribution of electrons
amongst the atomic orbitals. For example, for large values of $I_{E_{i}}$, the
number of atomic orbitals with non–zero probability of electron occupation is
very small, with a relatively large probability of occupation for each.
Moreover the corresponding vertices in the molecular graph are very
‘localised’, as there exists a substantial number of $E_{i}$–core–forbidden
vertices which are ‘distant’ from these $E_{i}$–core vertices. Further
interpretations are discussed in Remark 3.7, in a more general context.

For simplicity, we shall consider $\mathbf{H}=\mathbf{A}$. Singular graphs
have an important role in the Hückel framework, as the zero energy level
$E=0$, and its multiplicity, correspond to the non–bonding orbitals of the
molecule. The $0$–core vertices correspond to the atomic orbitals of which the
molecular orbitals for $E=0$ are a linear combination. Consider the $0$–CDP
for $\mathbf{U}=\mathbf{A}$. By Proposition 4.3, $I_{0}$ attains a minimum if
and only if $G$ is a slim graph. In particular bipartite MCs, which are a
sub–class of MCs and the building blocks of singular graphs, always have
$I_{0}$ at a minimum.

######  Example 4.5.

More generally, suppose we have two slim graphs $H_{1}=G_{1}+v_{1}$ and
$H_{2}=G_{2}+v_{2}$, where $v_{1}$ and $v_{2}$ are not cut vertices and are
both $0$–core vertices. Let $H_{1}$ have $n_{1}$ vertices and
$k_{1}=|CV_{0}(H_{1})|$. Similarly for $G_{2}+v_{2}$. Ali et al. [2016] showed
that the coalescence of $H_{1}$ and $H_{2}$ on $v_{1}$ and $v_{2}$ yields the
graph $H_{1}\circ H_{2}=G_{1}+G_{2}+v$, where $v$ is also a $0$–core vertex,
and the nullity is $\eta(H_{1})+\eta(H_{2})-1$. Clearly the $0$–core vertices
in $H_{1}$ which are not $v_{1}$ are preserved, and similarly for $H_{2}$.
Hence $|CV_{0}(H_{1}\circ H_{2})|=k_{1}+k_{2}-1$. It is also worth noting that
$H_{1}\circ H_{2}$ is also a slim graph and therefore $I_{0}(H_{1}\circ
H_{2})$ is at a minimum. Without loss of generality, let
$\frac{k_{1}}{n_{1}}<\frac{k_{2}-1}{n_{2}-1}$. Then
$\frac{k_{1}}{n_{1}}<\frac{k_{1}+k_{2}-1}{n_{1}+n_{2}-1}<\frac{k_{2}}{n_{2}}$.
It follows that $I_{0}(H_{1}\circ H_{2})$ is some value between $I_{0}(H_{1})$
and $I_{0}(H_{2})$, reflecting the ‘averaged’ distribution of $0$–core
vertices from $H_{1}$ and $H_{2}$ in $H_{1}\circ H_{2}$, both in probabilistic
terms for electron occupation, and in terms of the distribution of $0$–core
vertices within the molecular graph.

Future directions include the study of changes in $I_{0}$ as larger molecular
graphs are constructed, through eg. edge additions or graph coalescence
operations.

## Acknowledgement

The author is grateful to the anonymous referee for their comments, which have
led to an overall improvement in the presentation of this paper.

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