# Neutrino cooled disk in post-merger system studied via numerical GR MHD
simulation with a composition-dependent equation of state ††thanks: Presented
at the $8^{\rm th}$ Conference of the Polish Society on Relativity, Warsaw,
Poland, 19-23 September, 2022.

Agnieszka Janiuk
Center for Theoretical Physics of the Polish Academy of Sciences,
Al. Lotników 32/46, 02-668 Warsaw, Poland

###### Abstract

The code HARM_COOL, a conservative scheme for relativistic
magnetohydrodynamics, is being developed in our group and works with a
tabulated equation of state of dense matter. This EOS can be chosen and used
during dynamical simulation, instead of the simple ideal gas one. In this
case, the inversion scheme between the conserved and primitive variables is
not a trivial task. In principle, the code needs to solve numerically five
coupled non-linear equations at every time-step. The 5-D recovery schemes were
originally implemented in HARM and worked accurately for a simple polytropic
EOS which has an analytic form. Our current simulations support the
composition-dependent EOS, formulated in terms of rest-mass density,
temperature and electron fraction. In this proceeding, I discuss and compare
several recovery schemes that have been included in our code. I also present
and discuss their convergence tests. Finally, I show set of preliminary
results of a numerical simulation, addressed to the post-merger system formed
after the binary neutron stars (BNS) coalescence.

##  1 Introduction

The first detection of a gravitational wave signal accompanied by an
electromagnetic counterpart was announced on August 17, 2018, by the LIGO-
Virgo team [1]. The signal originated from the coalescence of two neutron
stars of masses in the range of 1.17-1.60 $M_{\odot}$ and the total mass of
the system of 2.74 $M_{\odot}$. It was followed by a short weak gamma ray
burst, observed 1.7 seconds after the GW signal. Hence, the theoretical
prediction that this class of short gamma ray bursts originate from compact
binary mergers has been proven by the detection of the source GW 170817.

It was also suggested before that the radioactivities from dynamical ejecta
after the first neutron star had been disrupted, can power an electromagnetic
signal [8]. Subsequent accretion onto a newly formed black hole can provide
bluer emission, if it is not absorbed by precedent ejecta [13, 7]. In this
case, a day-timescale emission comes at optical wavelengths from lanthanide-
free components of the ejecta, and is followed by a week-long emission with a
spectral peak in the near-infrared (NIR). This two component model fits well
with observations of the kilonova, detected in coincidence with the source
GRB-GW 170817.

In our studies, we investigate the very last stage of the system, namely the
post-merger black hole accretion disk. Due to its high density, the accretion
disk in post-merger system is opaque to photons. Neutrinos are produced via
$\beta$-reactions, electron-positron pair annihilation, and plasmon decay and
provide cooling mechanism. The plasma is composed of free nucleons, pairs, and
Helium. In the outer regions, heavier nuclei can be also synthesized, under
conditions of nuclear statistical equilibrium (NSE, see [5] for details). The
kilonova signal is produced in the equatorial wind outflows, launched from the
disk via magnetic instabilities. These ejecta are dominated by thermal energy
of the dense plasma, and are accelerated to mildly relativistic velocities (of
about 0.2-0.3 $c$, [6]). The matter is highly neutronized there, so due to the
r-process nucleosynthesis, copious amounts of unstable heavy isotopes are
formed in these winds, and power the Infrared/Optical emission through their
radioactive decay.

##  2 Numerical modeling

Our study of the winds launched from the accretion disk, is done by evolving
the general relativistic magnetohydrodynamic (GR MHD) equations in time. We
use the HARM (High Accuracy Relativistic Magnetohydrodynamics) code [4] which
is a conservative and shock capturing scheme. The numerical scheme advances
the conserved quantities from one time step to the next by solving a set of
non-linear hyperbolic equations for continuity, energy-momentum conservation
and magnetic induction. In the GR MHD scheme they read:

$\nabla_{\mu}(\rho u^{\mu})=0,\\\ \nabla_{\mu}(T^{\mu\nu})=0,\\\
\nabla_{\mu}(u^{\nu}b^{\mu}-u^{\mu}b^{\nu})=0$ (1)

where

$T^{\mu\nu}=T^{\mu\nu}_{\rm gas}+T^{\mu\nu}_{\rm EM}$ (2)

is contributed by

$T^{\mu\nu}_{\rm gas}=\rho
hu^{\mu}u^{\nu}+pg^{\mu\nu}=(\rho+u+p)u^{\mu}u^{\nu}+pg^{\mu\nu},$ (3)
$T^{\mu\nu}_{\rm
EM}=b^{2}u^{\mu}u^{\nu}+\frac{1}{2}b^{2}g^{\mu\nu}-b^{\mu}b^{\nu},~{}~{}~{}b^{\mu}=u^{*}_{\nu}F^{\mu\nu}.$
(4)

In the stress-energy tensor composed of the gas and electromagnetic terms,
$u^{\mu}$ is the four-velocity of the gas, $u$ is the internal energy, $\rho$
is the density, $p$ is the pressure, and $b^{\mu}$ is the magnetic four
vector. $F$ is the Faraday tensor and in a force-free approximation, we have
$E_{\nu}=u^{\nu}F^{\mu\nu}=0$. The unit convention is adopted such that
$G=c=M=1$.

Our initial conditions mimic the configuration after the transient (hyper-
massive-neutron star, HMNS) object has collapsed to a black hole, and a
pressure equilibrium torus [3] has been formed. In this solution, the angular
momentum along the radius of the disk is constant.

We parameterize our models with black hole Kerr parameter, $a$, and the inner
radius and radius of pressure maximum, $r_{\rm in}$, and $r_{\rm max}$,
respectively. The current simulations are run in an axisymmetric setup, i.e.
2D, with a resolution 256x256 points in the $r$, and $\theta$ directions. The
numerical code works in Kerr-Schild coordinates, which enables the matter to
smoothly accrete under the horizon.

The torus is embedded in an initially poloidal magnetic field, prescribed with
the vector potential of $(0,0,A_{\varphi})$, with
$A_{\varphi}=({\bar{\rho}\over\rho_{\rm max}}-\rho_{0})$ where we use offset
of $\rho_{0}=0.2$.

We parameterize the field strength with plasma $\beta$ defined as the ratio of
the gas-to-magnetic pressure, $\beta=p_{\rm gas}/p_{\rm mag}$. Here $p_{\rm
gas}=(\gamma-1)u_{\rm max}$ and $p_{\rm mag}=b_{\rm max}^{2}/2$, where $u_{\rm
max}$ is the internal energy of the gas at the radius of maximum pressure.

##  3 Chemical composition and structure of the disk

On eof significant chalelnges for numerical simulation is neutrino treatment.
Neutrinos carry away energy and lepton number, so they alter electron fraction
and composition of ejected material. Dynamical simulations must consider the
realistic equation of state (EOS) and impact of neutrinos in the optically
thin and thick regions. Lepton number conservation is expressed as follows:

$\nabla_{\mu}(n_{e}u^{\mu})={\mathcal{R}}/m_{b};~{}~{}m_{b}=\rho/n_{b};~{}~{}Y_{e}={n_{e}\over
n_{b}}={m_{b}n_{e}\over\rho}$ (5)

where ${\mathcal{R}}$ is the net rate of neutrino and antineutrino volume
number densities in the fluid frame, and $Y_{e}$ is the electron fraction.

Since the baryons dominate the rest-mass density, the baryon number
conservation equation turns to the regular continuity equation. In the energy-
momentum conservation equation, we must introduce an additional source term
due to heating and cooling by neutrinos:

$\nabla_{\mu}T^{\mu}_{\nu}={\mathcal{Q}}u_{\nu},$ (6)

where ${\mathcal{Q}}$ is the energy change per unit volume due to neutrino
emission.

###  3.1 Conserved and primitive variables

Conservation equations solved by the GR MHD code can be expressed in a flux
conservative form [14], and explicit choice of the conserved variables (which
are analytic functions of primitive ones) is to some extent arbitrary. The
recovery schemes for the primitive variables need numerical root finding, and
can be broadly devided into two categories. In a 2D scheme, there are two
independent variables, e.g, $v^{2}$ and specific enthalpy $W$. Temperature is
obtained from the EOS tables, solving $h=h(\rho,T,Y_{e})$, and pressure and
temperature are also obtained by Newton-Raphson method for $W$ and $v^{2}$.
Alternatively, the system of GR MHD equations is reduced to 3 equations, which
have three unknowns. The chosen independent variables can be: $\gamma$, $T$,
and $W=h\rho\gamma^{2}$. Pressure is interpolated from EOS tables, as
$P(\rho,T,Y_{e})$.

###  3.2 Recovery transformation

In the 2D recovery scheme of [9] the dimensionality of the recovery problem is
reduced by making use of certain scalar quantities that can be computed from
the conservatives. To avoid numerical pathologies o the $1D_{W}$ scheme near
the roots, this 2D scheme solves simultaneously the set of two equations:

$f_{1}:~{}~{}\tilde{Q}^{2}=v^{2}({\mathcal{B}}^{2}+W)^{2}-{{(Q_{\mu}{\mathcal{B}}^{\mu})^{2}({\mathcal{B}}^{2}+2W)}\over{W^{2}}}$
$f_{2}:~{}~{}Q_{\mu}n^{\mu}=-{{\mathcal{B}}^{2}\over
2}(1+v^{2})+{{(Q_{\mu}{\mathcal{B}}^{\mu})^{2}}\over{2W^{2}}}-W+p(u,\rho)$

The independent variables used in this scheme are defined as:
$Q_{\mu}=-n_{\mu}T^{\nu}_{\mu}=\alpha T^{t}_{\mu}$; where
$\tilde{Q}^{\nu}=j^{\nu}_{\mu}Q^{\mu}$ is energy-momentum density in the
normal obsever frame, and $D=-\rho n_{\mu}u^{\mu}=\alpha\rho
u^{t}=\gamma\rho$; is mass density in the observer’s frame, and
${\mathcal{B}}^{i}=\alpha B^{i}=\alpha{{}^{{}^{*}}\\!\\!F}^{it}$ is magnetic
3-vector. Here, $w=\rho+u+p$ with $W=w\gamma^{2}$ is enthalpy. To solve this
set of equations by means of Newton-Raphson method, the Jacobian matrix with
$\partial f_{1}\over\partial(v^{2})$, $\partial f_{2}\over\partial(v^{2})$,
$\partial f_{1}\over\partial W$, and $\partial f_{2}\over\partial W$ is
needed. Note that this scheme does not require an analytic EOS, and
derivatives of pressure wtr. to $\rho$, $v^{2}$, and $u$ may be computed from
tables using finite difference method.

In the 3D recovery scheme of [14], the system is extended to solve 3
equations, on $W,z$, and $T$, by adding a constraint on the internal energy
given by EOS tables:

$f_{1}:~{}~{}[\tau+D-z-B^{2}+{B^{i}S_{i}\over 2z^{2}}+\rho]W^{2}-{B^{2}\over
2}=0$

,

$f_{2}:~{}~{}[(z+B^{2})^{2}-S^{2}-{{2z+B^{2}}\over{z^{2}}}(B^{i}S_{i})^{2}]W^{2}-(z+B^{2})^{2}=0$

,

$f_{3}:~{}~{}\epsilon-\epsilon(\rho,T,Y_{e})=0$

. Here the temperature is employed directly as an unknown through
$\epsilon(W,zT)=h-1-{P\over\rho}={z-DW-\rho W^{2}\over DW}$ and does not
require inversion of the EOS. Notice here different notation: $S_{i}$ is
energy-momentum density, $W$ is Lorentz factor, $z$ is enthalpy, and
$\tau=-(n_{\mu}n_{\nu}T^{\mu\nu}+D)$.

Figure 1: Convergence test results for the 3D method (2 versions, with (1): we
compute specific internal energy from state vector x and conservatives as in
Eq. (25) in [2], and solve $f_{3}$ or (2): we compute pressure from state
vector x and conservatives and solve $f_{3}$ ), shown in the top panels, the
2D method (bottom panel, left) and Palenzuela method (bottom panel, right).

In method proposed in [12] the scheme is solving 1D equation, for the rescaled
variable, $\chi={{\rho h\gamma^{2}}\over{\rho\gamma}}$. Other quantities are
also rescaled, accordingly, to give Lorentz factor, and we give the brackets
for $\chi$: $2-2{{Q_{\mu}n^{\mu}+D}\over{D}}-{{\mathcal{B}}^{2}\over
D}<\chi<1-{{Q_{\mu}n^{\mu}+D}\over{D}}-{{\mathcal{B}}^{2}\over D}$. The
equation

$f(\chi)=\chi-\tilde{\gamma}(1+\tilde{\epsilon}+{\tilde{P}\over\tilde{\rho}})=0$

is solved, with $\tilde{P}=P(\tilde{\rho},\tilde{\epsilon},Y_{e})$ found in
tables.

The last method has proven to be most robust, for our wide parameter space. It
works with smaller errors, while the performance spped is also slower. We
performed the convergence tests to explore which recovery transformation works
best for the wide parameter space. The parameters used for testing routines
were: $Y_{e}=0.1$, $\gamma=2$, $p_{\rm gas}/p_{\rm mag}=10^{5}$. We derived
the conserved variables in Kerr metric, and then computed primitives perturbed
by a factor of 1.05. The variables recovered through the 2D scheme were
compared to the unperturbed, to calculate the total error, summed over all
primitives ${\rm Err}=\Sigma_{k=0,NPR}(P_{k}-\bar{P_{k}})^{2}$. Figure 1 shows
the results for all 3 methods probed.

###  3.3 Neutrino transport

We employ the neutrino leakage scheme that computes a gray optical depth
estimate along radial rays for electron neutrinos, electron antineutrinos, and
heavy-lepton neutrinos (nux), and then computes local energy and lepton number
loss terms. The source code of the scheme has been publicly available and
downloaded from https://stellarcollapse.org. Details are described e.g. in
[11]. The initial tests were done within an optically thin regime for
neutrinos.

$\tau(r,\theta,\phi)=\int_{r}^{R}\sqrt{\gamma_{rr}}\bar{\kappa}_{\nu_{i}}dr^{\prime}<2/3.$
(7)

##  4 Results

Figure 2: Maps of density and magnetic field contours (left), electron
fraction (middle) and neutrino emissivity (right panel). Model assumes
optically thin plasma for neutrinos. Inversion scheme used the Palenzuela
method. The model parameters are disk mass = 0.13 $M_{\odot}$, gas-to-magnetic
pressure ratio $\beta=50$ and black hole spin $a=0.6$. The color maps are in
logarithmic scale and are taken at $t=0.044$ s.

The structure of the flow in the evolved state is shown in Fig. 2. We present
polar slices, taken at the evolved time of the simulation ($t=44~{}ms$, which
is equivalent to the 3000 gometric time units, $t_{g}$, for black hole mass of
3 $M_{\odot}$). Parameters of this model are black hole spin $a=0.6$, initial
torus radius $r_{\rm in}=4r_{g}$ and pressure maximum radius of $r_{\rm
max}=11.8r_{g}$. We also assume initial electron fraction in the torus of
$Y_{\rm e,disk}=0.1$, and in the atmosphere it is $Y_{\rm e,atm}=0.45$. The
specific entropy per baryon in the torus is assumed $S=10k_{\rm B}$. This,
with the enthalpy value resulting from FM torus sulution, gives the physical
density scaling: units in the code are normalized such that $\rho_{\rm
max}=1$. The torus mass in cgs units is about $0.03M_{\odot}$.

As shown in the figure, the magnetic turbulence developed in the torus, and
helped launching winds from its surface. The neutronized material is
redistributed, and electron fraction in the winds became larger han in the
torus, about $Y_{e}=0.3$. The neutrino emissivity in the winds is couple of
orders of magnitude less than in the torus. As can be seen in the 2D map, the
accreting torus has a very high neutrino luminosity, but here neutrinos are
partially trapped. A moderately high neutrino luminosity can be observed from
the wind, where the neutrino-antineutrino pairs can contribute to the heating
of the plasma.

Neutronisation of the plasma in the tours is still significant, and electron
fraction values are below $Y_{e}=0.1-0.3$ in some densest regions (cf. [10]).
The neutrinos which are created in weak interactions, and the electron-
positron pair annihilation, nucleon bremsstrahlung and plasmon decay, provide
an efficient cooling mechanism. In the currently developed new numerical
scheme, we substituted a simplified description of the neutrino cooling rate
given by the two-stream approximation previously used by [6] with a more
advanced neutrino leakage scheme. The neutrino emissivity distribution image
taken at an evolved state of the torus is presented in Fig. 2 for the
optically thin case. For optically thick simulation (which assumed different
initial parameters as for the specific entropy), the quantitative difference
can be noticed. The neutrino and antineutrino luminosities, as a function of
time, are shown in Figure 3.

The velocity of the wind outflows, $v\sim(0.11-0.23)~{}c$, and mass loss via
unbound outflows of 2- 17% of the initial disk mass were determined previously
by [6]. We showed that the details are sensitive to engine parameters: BH spin
and magnetisation of the disk, namely the more magnetized disks produce faster
outflows. We also found that the accretion disk ejecta produce heavy elements
up to mass number $A\sim 200$, including platinium and gold isotopes (see
details in [10]).

Figure 3: The electron neutrino and antineutrino (Lnu1 and Lnu2, respectively;
Lnu3 denotes luminosity in other flavors) luminosty as a function of time, for
our two models, optically thin and optically thick, calculated with the
neutrino-leakage scheme.

##  5 Conclusions

Numerical GR MHD simulations have been widely used to model post-merger
systems and engines of short gamma ray bursts. We implemented a new scheme in
our HARM-COOL code to the case study of post-merger system and kilonova
source. The calculations are complex and need to cover physics of dense
nuclear matter. Their performance is sensitive to the chosen recovery schemes.
We tested several of them and chose the most robust for a lare parameter space
in densities, temperatures, and electron fraction values, to work with the
3-parameter EOS implemented in hydrodynamical simulation. Proper source terms
have been added in the system of equations for the neutrino losses, coupled
with composition changes. We also use an advanced neutrino-leakage scheme and
calculate emissivities of neutrino and antineutrino as a function of time.

The unbound outflows, i.e. winds, are powered by both neutrinos and
magnetically driven acceleration. Therefore, winds may be more dense and
powerful, if neutrino heating supports the magnetically-driven wind.

## Acknowledgment

This work was supported by grant 2019/35/B/ST9/04000 from the Polish National
Science Center. We used computational resources of the ICM of the Warsaw
University, and the PL-Grid, under grant plggrb5.

## References

  * [1] B. P. Abbott and others.  Gravitational Waves and Gamma-Rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A.  ApJL, 848(2):L13, 2017.
  * [2] P. Cerdá-Durán, J. A. Font, L. Antón, and E. Müller.  A new general relativistic magnetohydrodynamics code for dynamical spacetimes.  A&A, 492(3):937–953, December 2008.
  * [3] L. G. Fishbone and V. Moncrief.  Relativistic fluid disks in orbit around Kerr black holes.  ApJ, 207:962–976, 1976.
  * [4] C. F. Gammie, J. C. McKinney, and G. Tóth.  HARM: A Numerical Scheme for General Relativistic Magnetohydrodynamics.  ApJ, 589(1):444–457, 2003.
  * [5] A. Janiuk.  Nucleosynthesis of elements in gamma-ray burst engines.  A&A, 568:A105, 2014.
  * [6] A. Janiuk.  The r-process Nucleosynthesis in the Outflows from Short GRB Accretion Disks.  ApJ, 882(2):163, 2019.
  * [7] O. Korobkin, S. Rosswog, A. Arcones, and C. Winteler.  On the astrophysical robustness of the neutron star merger r-process.  MNRAS, 426(3):1940–1949, 2012.
  * [8] L.-X. Li and B. Paczyński.  Transient Events from Neutron Star Mergers.  ApJL, 507(1):L59–L62, 1998.
  * [9] S. C. Noble, C. F. Gammie, J. C. McKinney, and L. Del Zanna.  Primitive Variable Solvers for Conservative General Relativistic Magnetohydrodynamics.  ApJ, 641(1):626–637, 2006.
  * [10] F. H. Nouri, A. Janiuk, and M. Przerwa.  Studying Postmerger Outflows from Magnetized-neutrino-cooled Accretion Disks.  ApJ, 944(2):220, 2023.
  * [11] C. D. Ott, E. Abdikamalov, E. O’Connor, C. Reisswig, R. Haas, P. Kalmus, S. Drasco, A. Burrows, and E. Schnetter.  Correlated gravitational wave and neutrino signals from general-relativistic rapidly rotating iron core collapse.  Phys.Rev.D, 86(2):024026, 2012.
  * [12] C. Palenzuela, S. L. Liebling, D. Neilsen, L. Lehner, O. L. Caballero, E. O’Connor, and M. Anderson.  Effects of the microphysical equation of state in the mergers of magnetized neutron stars with neutrino cooling.  Phys.Rev.D, 92(4):044045, 2015.
  * [13] D. M. Siegel and B. D. Metzger.  Three-Dimensional General-Relativistic Magnetohydrodynamic Simulations of Remnant Accretion Disks from Neutron Star Mergers: Outflows and r -Process Nucleosynthesis.  PRL, 119(23):231102, 2017.
  * [14] D. M. Siegel, P. Mösta, D. Desai, and S. Wu.  Recovery Schemes for Primitive Variables in General-relativistic Magnetohydrodynamics.  ApJ, 859(1):71, 2018.


# On $A_{\alpha}$-spectrum of joined union of graphs and its applications to
power graphs of finite groups

Bilal A. Rathera, Hilal A. Ganieb, S. Pirzadac
a,cDepartment of Mathematics, University of Kashmir, Srinagar, India
b Department of School Education, JK Govt. Kashmir, India
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Abstract. For a simple graph $G$, the generalized adjacency matrix
$A_{\alpha}(G)$ is defined as $A_{\alpha}(G)=\alpha
D(G)+(1-\alpha)A(G),\alpha\in[0,1]$, where $A(G)$ is the adjacency matrix and
$D(G)$ is the diagonal matrix of vertex degrees of $G$. This matrix
generalises the spectral theories of the adjacency matrix and the signless
Laplacian matrix of $G$. In this paper, we find $A_{\alpha}$-spectrum of the
joined union of graphs in terms of spectrum of adjacency matrices of its
components and the zeros of the characteristic polynomials of an auxiliary
matrix determined by the joined union. We determine the $A_{\alpha}$-spectrum
of join of two regular graphs, the join of a regular graph with the union of
two regular graphs of distinct degrees. As an applications, we investigate the
$A_{\alpha}$-spectrum of certain power graphs of finite groups.

Keywords: Adjacency matrix, signless Laplacian matrix, $A_{\alpha}$-matrix,
power graphs, finite groups, totient function

AMS subject classification: 05C50, 05C12, 15A18.

##  1 Introduction

Let $G$ be a simple graph with $n$ vertices and $m$ edges having vertex
$V(G)=\\{v_{1},v_{2},\dots,v_{n}\\}$ and edge set
$E(G)=\\{e_{1},e_{2},\dots,e_{m}\\}$. The set of vertices adjacent to a vertex
$v\in V(G)$, denoted by $N(v)$, refers to the neighborhood of $v.$ The degree
of a vertex $v,$ denoted by $d_{G}(v)$ (we simply write $d_{v}$) is the
cardinality of $N(v)$. A graph is called regular if each of its vertices have
the same degree. We write $v_{i}\sim v_{j}$, if $v_{i}$ and $v_{j}$ are
connected by an edge, otherwise we write $v_{i}\nsim v_{j}.$ The adjacency
matrix $A=(a_{ij})$ of $G$ is a $(0,1)$-square matrix of order $n$ whose
$(i,j)$-entry is equal to $1$, if $v_{i}\sim v_{j}$ and equal to $0$,
otherwise. Let $D(G)=diag(d_{1},d_{2},\dots,d_{n})$ be the diagonal matrix of
vertex degrees $d_{i}=d_{G}(v_{i}),i=1,2,\dots,n$ of $G.$ The matrices
$L(G)=A(G)-D(G)$ and $Q(G)=A(G)+D(G)$ are called the Laplacian matrix and the
signless Laplacian matrix, respectively of the graph $G$. Both $L(G)$ and
$Q(G)$ are positive semidefinite matrices having real eigenvalues so that
their eigenvalues can be put as
$\mu_{1}(G)\geq\mu_{2}(G)\geq\dots\geq\mu_{n}(G)=0$ and $q_{1}(G)\geq
q_{2}(G)\geq\dots\geq q_{n}(G)$, respectively.

Nikiforov [17] proposed to study the convex combinations $A_{\alpha}(G)$ of
$A(G)$ and $D(G)$ defined by $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$,
$0\leq\alpha\leq 1$. Clearly, $A(G)=A_{0}(G)$, $D(G)=A_{1}(G)$ and
$2A_{\frac{1}{2}}(G)=D(G)+A(G)=Q(G)$. We further note that
$A_{\alpha}(G)-A_{\gamma}(G)=(\alpha-\gamma)(D(G)-A(G))=(\alpha-\gamma)L(G)$,
where $L(G)$ is the Laplacian matrix of $G$. Therefore, the matrix
$A_{\alpha}(G)$ is a generalisation of the adjacency matrix $A(G)$ and the
signless Laplacian matrix $Q(G)$ of $G$. Let $A(G)$ be the adjacency matrix of
the graph $G$ and let $B$ be a real diagonal matrix of order $n$. Bapat et al.
[2] defined the matrix $L^{{}^{\prime}}=B-A(G)$ and called it the perturbed
Laplacian matrix of $G$. For $\alpha\neq 1$, it is easy to see that

$\displaystyle A_{\alpha}(G)=\alpha
D(G)+(1-\alpha)A(G)=(\alpha-1)\Big{(}\frac{\alpha}{\alpha-1}D(G)-A(G)\Big{)}.$

Clearly $\frac{\alpha}{\alpha-1}D(G)$ is a diagonal matrix with real entries,
so that $A_{\alpha}(G)$ is a scaler multiple of a perturbed Laplacian matrix.
This is another motivation to study the spectral properties of
$A_{\alpha}(G)$. For some recent papers on the spectral properties of
$A_{\alpha}(G)$, we refer to [17, 18, 14, 24, 22] and the references therein.

The matrix $A_{\alpha}(G)$ is a real symmetric matrix, therefore we can
arrange its eigenvalues as
$\lambda_{1}(A_{\alpha}(G))\geq\lambda_{2}(A_{\alpha}(G))\geq\cdots\geq\lambda_{n}(A_{\alpha}(G)),$
where $\lambda_{1}(A_{\alpha}(G))$ is called the generalized adjacency
spectral radius of $G$. For a connected graph $G$, the matrix $A_{\alpha}(G)$
is non-negative and irreducible (for $\alpha\neq 1$). Therefore, by Perron-
Frobenius theorem, $\lambda(A_{\alpha}(G))$ is the simple (with multiplicity
$1$) eigenvalue and there is a unique positive unit eigenvector $X$
corresponding to $\lambda(A_{\alpha}(G)),$ which is called the generalized
adjacency Perron vector of $G.$

As usual $K_{n},~{}K_{1,n-1}~{}P_{n}\text{and}~{}C_{n}$ respectively, denote
the complete graph, the star, the path and the cycle graph of order $n$. For
other notations and terminology we refer to [9, 12, 21, 19].

The rest of the paper is organized as follows. In Section 2, we obtain the
$A_{\alpha}$-spectrum of the joined union of regular graphs
$G_{1},G_{2},\dots,G_{n}$ in terms of their adjacency spectrum and the
spectrum of an auxiliary matrix. As a consequence, we obtain
$A_{\alpha}$-spectrum of some well known graphs. In Section 3, we investigate
the $A_{\alpha}$-spectrum of the power graphs of some finite groups and raise
some problems for further work.

##  2 $A_{\alpha}$-spectrum of the joined union of graphs

Consider an $n\times n$ matrix

$M=\begin{pmatrix}A_{1,1}&A_{1,2}&\cdots&A_{1,s}\\\
A_{2,1}&A_{2,2}&\cdots&A_{2,s}\\\ \vdots&\vdots&\cdots&\vdots\\\
A_{s,1}&A_{s,2}&\cdots&A_{s,s}\\\ \end{pmatrix},$

whose rows and columns are partitioned according to a partition
$P=\\{P_{1},P_{2},\dots,P_{m}\\}$ of the set $X=\\{1,2,\dots,n\\}.$ The
quotient matrix $\mathcal{Q}$ (see [4, 9]) is an $s\times s$ matrix whose
entries are the average row sums of the blocks $A_{i,j}$ of $M$. The partition
$P$ is said to be equitable if each block $A_{i,j}$ of $M$ has constant row
(and column) sum and in this case the matrix $\mathcal{Q}$ is called as
equitable quotient matrix. A vertex partition $\\{V_{1},V_{2},\dots,V_{s}\\}$
of the vertex set $V(G)$ of $G$ is equitable if for each $i$ and for all
$u,v\in V_{i}$, we have $|N(u)\cap V_{j}|=|N(v)\cap V_{j}|$, for all $j.$ In
general, the eigenvalues of $\mathcal{Q}$ interlace the eigenvalues of $M$. In
case the partition is equitable, the following lemma was obtained in [4].

###### Lemma 2.1

[4, 9] If the partition $P$ of $X$ of matrix $M$ is equitable, then each
eigenvalue of $\mathcal{Q}$ is an eigenvalue of $M.$

Consider a graph $G$ having symmetry so that its associated matrix is written
in the form

$\displaystyle M=\begin{pmatrix}X&\beta&\cdots&\beta&\beta\\\
\beta^{T}&B&\cdots&C&C\\\ \vdots&\vdots&\cdots&\vdots&\vdots\\\
\beta^{T}&C&\cdots&B&C\\\ \beta^{T}&C&\cdots&C&B\end{pmatrix},$ (2.1)

where $X\in R^{t\times t},~{}\beta\in R^{t\times s}$ and $B,C\in R^{s\times
s}$, such that $n=t+cs$, with $c$ being the number of copies of $B$. The
spectrum of this matrix can be obtained as the union of the spectrum of
smaller matrices using the following technique given in [10]. In the statement
of the theorem, $\sigma^{(k)}(Y)$ indicates the multi-set formed by $k$ copies
of the spectrum of $Y$, denoted by $\sigma(Y)$.

###### Lemma 2.2

Let $M$ be a matrix of the form given in (2.1), with $c\geq 1$ copies of the
block $B$. Then

  1. (i)

$\sigma(B-C)\subseteq\sigma(M)$ with multiplicity $c-1$;

  2. (ii)

$\sigma(M)\setminus\sigma^{(c-1)}(B-C)=\sigma(M^{\prime})$ is the set of the
remaining $t+s$ eigenvalues of $M$, where
$M^{\prime}=\begin{pmatrix}X&\sqrt{c}\beta\\\
\sqrt{c}\beta^{T}&B+(c-1)C\end{pmatrix}.$

Let $G(V,E)$ be a graph of order $n$ and $G_{i}(V_{i},E_{i})$ be graphs of
order $n_{i},$ where $i=1,\ldots,n$. The joined union $G[G_{1},\ldots,G_{n}]$
is the graph $H(W,F)$ with

$\displaystyle
W=\bigcup_{i=1}^{n}V_{i}~{}~{}~{}\text{and}~{}~{}F=\bigcup_{i=1}^{n}E_{i}\cup\bigcup_{\\{v_{i},v_{j}\\}\in
E}V_{i}\times V_{j}.$

In other words, the joined union is the union of graphs $G_{1},\ldots,G_{n}$
together with the edges $v_{ik}v_{jl},~{}v_{ik}\in G_{i}$ and $v_{jl}\in
G_{j}$, whenever $v_{i}v_{j}$ is an edge in $G$. Clearly, the usual join of
two graphs $G_{1}$ and $G_{2}$ is a special case of the joined union
$K_{2}[G_{1},G_{2}]=G_{1}\triangledown G_{2}$ where $K_{2}$ is the complete
graph of order $2$. From the definition, it is clear that the diameter of the
graph $G[G_{1},\ldots,G_{n}]$ is same as the diameter of the graph $G$, for
every choice of the connected graphs $G_{1},G_{2},\dots,G_{n}$, $n\geq 3$ and
$G\ncong K_{n}$.

Now, we obtain the $A_{\alpha}$-spectrum of the joined union of regular graphs
$G_{1},G_{2},\dots,G_{n}$ in terms of the adjacency spectrum of the graphs
$G_{1},G_{2},\dots,G_{n}$ and the eigenvalues of an auxiliary matrix.

###### Theorem 2.3

Let $G$ be a graph of order $n\geq 3$ having $m$ edges. Let $G_{i}$ be
$r_{i}$-regular graphs of order $n_{i}$ having adjacency eigenvalues
$\lambda_{i1}=r_{i}\geq\lambda_{i2}\geq\ldots\geq\lambda_{in_{i}},$ where
$i=1,2,\ldots,n$. The $A_{\alpha}$-spectrum of the joined union graph
$G[G_{1},\ldots,G_{n}]$ of order $N=\sum\limits_{i=1}^{n}n_{i}$ consists of
the eigenvalues $\alpha(r_{i}+\alpha_{i})+(1-\alpha)\lambda_{ik}(G_{i})$ for
$i=1,\ldots,n$ and $k=2,3,\ldots,n_{i}$, where
$\alpha_{i}=\sum\limits_{v_{j}\in N_{G}(v_{i})}n_{i}$ is the sum of the
cardinalities of the graphs $G_{j},j\neq i$ which corresponds to the
neighbours of vertex $v_{i}\in G$. The remaining $n$ eigenvalues are given by
the matrix

$M=\begin{pmatrix}\psi_{11}&(1-\alpha)\psi_{12}&\ldots&(1-\alpha)\psi_{1n}\\\
(1-\alpha)\psi_{21}&\psi_{22}&\ldots&(1-\alpha)\psi_{2n}\\\
\vdots&\vdots&\ddots&\vdots\\\
(1-\alpha)\psi_{n1}&(1-\alpha)\psi_{n2}&\ldots&\psi_{nn}\end{pmatrix},$

where $\psi_{ii}=\alpha\alpha_{i}+r_{i}$, for $i=1,2,\dots,n$ and for $i\neq
j$, $\psi_{ij}=n_{j}$, if $v_{i}\sim v_{j}$, while as $\psi_{ij}=0$, if
$v_{i}\nsim v_{j}$.

Proof. Let $V(G)=\\{v_{1},\ldots,v_{n}\\}$ be the vertex set of $G$ and let
$V(G_{i})=\\{v_{i1},\ldots,v_{in_{i}}\\}$ be the vertex set of $G_{i}$, for
$i=1,2,\dots,n$. Let $H=G[G_{1},\ldots,G_{n}]$ be the joined union of the
graphs $G_{1},G_{2},\dots,G_{n}$. Clearly, order of $H$ is
$N=\sum\limits_{i=1}^{n}n_{i}$. We first compute the degree of each vertex in
$V(H)$. Evidently, the degree of each vertex $v_{ij}\in V(H)$, for $1\leq
i\leq n$ and $1\leq j\leq n_{i}$, is the degree inside $G_{i}$ plus the sum of
cardinalities of the copies of $G_{j},j\neq i$ which corresponds to the
neighbours of the vertex $v_{i}$ in $G$. That is, for each $v_{ij}\in
V(G_{i})$, we have

$d_{H}(v_{ij})=r_{i}+\sum\limits_{v_{j}\in
N_{G}(v_{i})}n_{j}=r_{i}+\alpha_{i},$ (2.2)

where $\alpha_{i}=\sum\limits_{v_{j}\in N_{G}(v_{i})}n_{j}$. Obviously,
$d_{H}(v_{ij})$ is same for each vertex in $G_{i}$, $1\leq j\leq n_{i}$. Label
the vertices in $H$ from the vertices in $G_{1}$ to the vertices in $G_{n}$.
With this labelling, the $A_{\alpha}$-matrix of $H$ can be written as

$A_{\alpha}(H)=\begin{pmatrix}h_{1}&(1-\alpha)a(v_{1},v_{2})&\ldots&(1-\alpha)a(v_{1},v_{n})\\\
(1-\alpha)a(v_{2},v_{1})&h_{2}&\ldots&(1-\alpha)a(v_{2},v_{n})\\\
\vdots&\vdots&\ddots&\vdots\\\
(1-\alpha)a(v_{n},v_{1})&(1-\alpha)a(v_{n},v_{2})&\ldots&h_{n}\end{pmatrix},$

where,

$h_{i}=\alpha(r_{i}+\alpha_{i})I_{n_{i}}+(1-\alpha)A(G_{i}),\qquad\text{for}~{}i=1,2,\ldots,n,$

and $a(v_{i},v_{j})=J_{n_{i}\times n_{j}}$, if $v_{i}\sim v_{j}$ in $G$ and
$0_{n_{i}\times n_{j}}$, otherwise. For $i=1,2,\ldots,n,A(G_{i})$ is the
adjacency matrix of $G_{i}$, $J_{n_{i}\times n_{j}}$ is the matrix having all
entries $1$, $0_{n_{i}\times n_{j}}$ is the zero matrix of order $n_{i}\times
n_{j}$ and $I_{n_{i}}$ is the identity matrix of order $n_{i}$.
As $G_{i}$ is an $r_{i}$ regular graph, so the all one vector
$e_{n_{i}}=(1,1,\dots,1)^{T}$ with $n_{i}$ entries is the eigenvector of the
adjacency matrix $A(G_{i})$ corresponding to the eigenvalue $r_{i}$ and all
other eigenvectors are orthogonal to $e_{n_{i}}.$ Let $\lambda_{ik}$, $2\leq
k\leq n_{i}$, be any eigenvalue of $A(G_{i})$ with the corresponding
eigenvector $X=(x_{i1},x_{i2},\dots,x_{in_{i}})^{T}$ satisfying
$e_{n_{i}}^{T}X=0.$ Clearly, the column vector $X$ can be regarded as a
function defined on $V(G_{i})$ assigning the vertex $v_{ij}$ to $x_{ij}$, that
is, $X(v_{ij})=x_{ij}$ for $i=1,2,\ldots,n$ and $j=1,2,\dots,n_{i}$. Now,
consider the vector $Y=(y_{1},y_{2},\dots,y_{n})^{T}$, where

$y_{j}=\left\\{\begin{array}[]{rl}x_{ij}&~{}\text{if}~{}v_{ij}\in V(G_{i})\\\
0&~{}\text{otherwise.}\\\ \end{array}\right.$

As $e_{n_{i}}^{T}X=0$ and coordinates of the vector $Y$ corresponding to
vertices in $\cup_{j\neq i}V_{j}$ of $H$ are zeros, we have

$\displaystyle A_{\alpha}(H)Y=\begin{pmatrix}0\\\ \vdots\\\ 0\\\
\alpha(r_{i}+\alpha_{i})X+(1-\alpha)\lambda_{ik}X\\\ 0\\\ \vdots\\\
0\end{pmatrix}=\Big{(}\alpha(r_{i}+\alpha_{i})+(1-\alpha)\lambda_{ik}\Big{)}Y.$

This shows that $Y$ is an eigenvector of $A_{\alpha}(H)$ corresponding to the
eigenvalue $\alpha(r_{i}+\alpha_{i})+(1-\alpha)\lambda_{ik}$, for every
eigenvalue $\lambda_{ik}$, $2\leq k\leq n_{i}$, of $A(G_{i})$. From this it
follows that for $1\leq i\leq n$ and $2\leq k\leq n_{i}$,
$\alpha(r_{i}+\alpha_{i})+(1-\alpha)\lambda_{ik}$, is an eigenvalue of
$A_{\alpha}(H)$. So, we have obtain $\sum\limits_{i=1}^{n}n_{i}-n=N-n$
eigenvalues of $A_{\alpha}(H)$ with this procedure. The remaining $n$
eigenvalues are the zeros of the characteristic polynomial of the following
equitable quotient matrix.

$M=\begin{pmatrix}\phi_{11}&(1-\alpha)\phi_{12}&\ldots&(1-\alpha)\phi_{1n}\\\
(1-\alpha)\phi_{21}&\phi_{22}&\ldots&(1-\alpha)\phi_{2n}\\\
\vdots&\vdots&\ddots&\vdots\\\
(1-\alpha)\phi_{n1}&(1-\alpha)\phi_{n2}&\ldots&\phi_{nn}\end{pmatrix},$

where $\phi_{ii}=\alpha\alpha_{i}+r_{i}$, for $i=1,2,\dots,n$ and for $i\neq
j$, $\phi_{ij}=n_{j}$, if $v_{i}\sim v_{j}$, while as $\phi_{ij}=0$, if
$v_{i}\nsim v_{j}$. This completes the proof.

If in particular we take $G_{i}=K_{p_{i}}$ the complete graph on $p_{i}$
vertices (a ($p_{i}-1$)-regular graph), then using the fact that the adjacency
eigenvalues of $K_{p_{i}}$ are $p_{i}-1$ with multiplicity $1$ and the
eigenvalue $-1$ with multiplicity $p_{i}-1$, we have the following consequence
of Theorem 2.3.

###### Corollary 2.4

Let $G$ be a graph of order $n\geq 3$. For $i=1,2,\dots,n$, let
$G_{i}=K_{p_{i}}$. Then the $A_{\alpha}$-spectrum of
$G[K_{p_{1}},\ldots,K_{p_{n}}]$ consists of $\alpha p_{i}-1+\alpha\alpha_{i}$,
with multiplicity $p_{i}-1$, for each $i=1,2,\dots,n$, where
$\alpha_{i}=\sum\limits_{v_{j}\in N_{G}(v_{i})}n_{j}$. The remaining $n$
eigenvalues are given by the matrix

$\begin{pmatrix}\phi_{11}&(1-\alpha)\phi_{12}&\ldots&(1-\alpha)\phi_{1n}\\\
(1-\alpha)\phi_{21}&\phi_{22}&\ldots&(1-\alpha)\phi_{2n}\\\
\vdots&\vdots&\ddots&\vdots\\\
(1-\alpha)\phi_{n1}&(1-\alpha)\phi_{n2}&\ldots&\phi_{nn}\end{pmatrix},$

where $\phi_{ii}=\alpha\alpha_{i}+p_{i}-1$, for $i=1,2,\dots,n$ and for $i\neq
j$, $\phi_{ij}=n_{j}$, if $v_{i}\sim v_{j}$, while as $\phi_{ij}=0$, if
$v_{i}\nsim v_{j}$.

We observe that the complete $p$-partite graph $K_{n_{1},n_{2},\dots,n_{p}}$,
is the joined union of graphs $G_{i}=\overline{K}_{n_{i}}$, where underlying
graph is $K_{p}$. So,
$K_{n_{1},n_{2},\dots,n_{p}}=K_{p}[\overline{K}_{n_{1}},\overline{K}_{n_{2}},\dots,\overline{K}_{n_{p}}].$
The next consequence of Theorem 2.3 gives the $A_{\alpha}$-spectrum of the
complete $p$-partite graph $K_{n_{1},n_{2},\dots,n_{p}}$.

###### Corollary 2.5

The $A_{\alpha}$-spectrum of the complete $p$-partite graph
$K_{n_{1},n_{2},\dots,n_{p}}$ with $N=\displaystyle\sum_{i=1}^{p}n_{i}$
consists of the eigenvalue $\alpha(N-n_{i})$, for $i=1,2,\dots,p$, each having
multiplicity $n_{i}-1$ and the remaining $p$ eigenvalues are given by the
matrix

$\begin{pmatrix}\alpha(N-n_{1})&n_{2}(1-\alpha)&\ldots&n_{p}(1-\alpha)\\\
n_{1}(1-\alpha)&\alpha(N-n_{2})&\ldots&n_{p}(1-\alpha)\\\
\vdots&\vdots&\ddots&\vdots\\\
n_{1}(1-\alpha)&n_{2}(1-\alpha)&\ldots&\alpha(N-n_{p})\end{pmatrix}.$

Proof. This follows from Theorem 2.3 by taking $G=K_{p}$,
$G_{i}=\overline{K}_{n_{i}}$, $a(v_{i},v_{j})=J_{n_{i}\times n_{j}}$, for all
$i,j$, $\alpha_{i}=\sum\limits_{v_{j}\in N(v_{i})}n_{i}=N-n_{i},r_{i}=0$ and
$\lambda_{ik}=0$, $1\leq k\leq n_{i}-1$ for $1\leq i\leq p.$

###### Corollary 2.6

Let $G=K_{t,t,\dots,t}$ be a complete $p$-partite graph with $N=pt.$ The
$A_{\alpha}$-spectrum of $G$ consists of the eigenvalue $\alpha t(p-1)$ with
multiplicity $tp-p$, the eigenvalue $t(\alpha p-1)$ with multiplicity $p-1$
and the eigenvalue $t(p-1)$ with multiplicity $1$.

Proof. By Corollary 2.5, we see that $\alpha t(p-1)$ is an eigenvalue with
multiplicity $pt-p$ and other eigenvalues are given by

$\begin{pmatrix}\alpha t(p-1)&t(1-\alpha)&\ldots&t(1-\alpha)\\\
t(1-\alpha)&\alpha t(p-1)&\ldots&t(1-\alpha)\\\ \vdots&\vdots&\ddots&\vdots\\\
t(1-\alpha)&t(1-\alpha)&\ldots&\alpha t(p-1)\end{pmatrix}.$

This matrix has symmetry of the type (2.1), and so by taking $X$ and $\beta$
as empty matrices, $B=[\alpha t(p-1)]$ and $C=[t(1-\alpha)]$ in Lemma 2.1, we
get the eigenvalue $t(\alpha p-1)$ with multiplicity $p-1$. Finally
$B+(c-1)C=\alpha t(p-1)+(p-1)t(1-\alpha)=t(p-1)$ is the remaining eigenvalue.

The following observation gives the $A_{\alpha}$-spectrum of the join of a
regular graph with the union of two regular graphs of distinct degrees.

###### Corollary 2.7

For $i=1,2,3$, let $G_{i}$ be $r_{i}$ regular graphs of orders $n_{i}$ having
adjacency eigenvalues
$\lambda_{i1}=r_{i}\geq\lambda_{i2}\geq\dots\geq\lambda_{in_{i}}$. Then the
$A_{\alpha}$-spectrum of $G=G_{1}\triangledown(G_{2}\cup G_{3})$ consists of
the eigenvalues $\alpha(r_{1}+n-n_{1})+(1-\alpha)\lambda_{1k}(A(G_{1}))$,
$k=2,\dots,n_{1}$, the eigenvalues
$\alpha(n_{1}+r_{2})+(1-\alpha)\lambda_{2k}(A(G_{2}))$, $k=2,\dots,n_{2}$, the
eigenvalues $\alpha(n_{1}+r_{3})+(1-\alpha)\lambda_{3k}(A(G_{3}))$,
$k=2,\dots,n_{3}$, where $n=n_{1}+n_{2}+n_{3}$. The remaining three
eigenvalues are given by the matrix

$\begin{pmatrix}r_{2}+\alpha n_{1}&(1-\alpha)n_{1}&0\\\
(1-\alpha)n_{2}&r_{1}+\alpha(n_{2}+n_{3})&(1-\alpha)n_{3}\\\
0&(1-\alpha)n_{1}&r_{3}+\alpha n_{1}\end{pmatrix}.$

Proof. Let $P_{3}$ be the path on $3$ vertices and let
$G=G_{1}\triangledown(G_{2}\cup G_{3})$ be the join of the graphs $G_{1}$ and
$G_{2}\cup G_{3}$. It is easy to see that $G=P_{3}[G_{2},G_{1},G_{3}]$, that
is, $G$ is the joined union of the graphs $G_{2},G_{1},G_{3}$ when the parent
graph is the path $P_{3}$. Now, using Theorem 2.3 and noting that
$\alpha_{2}=n_{1}$, $\alpha_{1}=n_{2}+n_{3}$ and $\alpha_{3}=n_{1}$, the
result follows.

Let $G_{1}$ and $G_{2}$ be two graphs of order $n_{1}$ and $n_{2}$,
respectively. Let $G=G_{1}\triangledown G_{2}$ be the join of the graphs
$G_{1}$ and $G_{2}$. Clearly, $G=K_{2}[G_{1},G_{2}]$, that is, $G$ is the
joined union of $G_{1}$ and $G_{2}$ when the parent graph is $K_{2}$. The
following consequence of Theorem 2.3, gives the $A_{\alpha}$-spectrum of the
join of two regular graphs.

###### Corollary 2.8

Let $G_{i}$ be an $r_{i}$ regular graph of order $n_{i}$ for $i=1,2$. Let
$\lambda_{ik},2\leq k\leq n_{i},i=1,2$ be the adjacency eigenvalues of
$G_{i}$. Then the $A_{\alpha}$-spectrum of $G=G_{1}\triangledown G_{2}$
consists of eigenvalues $\alpha(r_{1}+n_{2})+(1-\alpha)\lambda_{1k}A(G_{1})$,
$k=2,\dots,n_{1}$, the eigenvalues
$\alpha(r_{2}+n_{1})+(1-\alpha)\lambda_{2k}A(G_{1}),k=2,\dots,n_{2}$ and the
remaining two eigenvalues are given by the matrix

$\begin{pmatrix}r_{1}+\alpha n_{2}&(1-\alpha)n_{2}\\\
(1-\alpha)n_{1}&r_{2}+\alpha n_{1}\end{pmatrix}.$ (2.3)

A _friendship_ graph $F_{n}$ is a graph of order $2n+1$, obtained by joining
$K_{1}$ with $n$ copies of $K_{2}$, that is,
$F_{n}=K_{1}\triangledown(nK_{2})=K_{1,n}[K_{1},\underbrace{K_{2},K_{2},\dots,K_{2}}_{n}]$,
where $K_{1}$ corresponds to the root vertex(vertex of degree greater than
one) in $K_{1,n}$. In particular, replacing some of $K_{2}$’s by $K_{1}$’s, in
$F_{n}$ we get _firefly_ type graph denoted by $F_{p,n-p}$ and written as

$F_{p,n-p}=K_{1,n}[K_{1},\underbrace{K_{1},K_{1},\dots,K_{1}}_{p}\underbrace{K_{2},K_{2},\dots,K_{2}}_{n-p}].$

The $A_{\alpha}$-spectrum of the _friendship_ graph $F_{n}$ and the _firefly_
type graph $F_{p,n-p}$ are given by following corollary.

###### Corollary 2.9

  * (i)

The $A_{\alpha}$-spectrum of $F_{n}$ consists of the eigenvalue $3\alpha-1$
with multiplicity $n$, the remaining eigenvalues are the zeros of the matrix

$\begin{pmatrix}2\alpha n&2(1-\alpha)&\dots&2(1-\alpha)\\\
1-\alpha&1+\alpha&\dots&2(1-\alpha)\\\ \vdots&\vdots&\vdots&\vdots\\\
1-\alpha&2(1-\alpha)&\dots&1+\alpha\end{pmatrix}.$

  * (ii)

The $A_{\alpha}$-spectrum of $F_{p,n-p}$ consists of the eigenvalue
$3\alpha-1$ with multiplicity $n-p$ and the eigenvalues of the matrix

$\begin{pmatrix}\alpha(2n-p)&1-\alpha&\dots&2(1-\alpha)&2(1-\alpha)&\dots&2(1-\alpha)\\\
1-\alpha&\alpha&\dots&1-\alpha&2(1-\alpha)&\dots&2(1-\alpha)\\\
\vdots&\vdots&\dots&\vdots&\vdots&\dots&\vdots\\\
1-\alpha&1-\alpha&\dots&\alpha&2(1-\alpha)&\dots&2(1-\alpha)\\\
1-\alpha&1-\alpha&\dots&1-\alpha&1+\alpha&\dots&2(1-\alpha)\\\
\vdots&\vdots&\dots&\vdots&\vdots&\dots&\vdots\\\
1-\alpha&1-\alpha&\dots&1-\alpha&2(1-\alpha)&\dots&1+\alpha\\\ \end{pmatrix}.$

Let $K_{a,b}$ be the complete bipartite graph with partite sets of cardinality
$a$ and $b.$ Clearly $K_{a,b}=\overline{K}_{a}\triangledown\overline{K}_{b}$.
A _complete split graph_ , denoted by $CS_{\omega,n-\omega}$, is the graph
consisting of a clique on $\omega$ vertices and an independent set (a subset
of vertices of a graph is said to be an independent set if the subgraph
induced by them is an empty graph) on the remaining $n-\omega$ vertices, such
that each vertex of the clique is adjacent to every vertex of the independent
set. Further, note that
$CS_{\omega,n-\omega}=K_{\omega}\triangledown\overline{K}_{n-\omega}$.
Similarly, the _cone graph_ is given by
$C_{a,b}=C_{a}\triangledown\overline{K}_{b}$ and for $b=1$, we have _wheel
graph_ $W_{n+1}=C_{n}\triangledown K_{1}$ on $n+1$ vertices. As a special case
of Corollary 2.8, we compute the $A_{\alpha}$-spectrum of the complete
bipartite graph $K_{a,b}$, the complete split graph $CS_{\omega,n-\omega},$
the cone graph $C_{a,b}$ and $W_{n+1}$ in the following corollary.

###### Corollary 2.10

  * (i)

The $A_{\alpha}$-spectrum of $K_{a,b}$ consists of the eigenvalue $\alpha b$
with multiplicity $a-1$, the eigenvalue $\alpha a$ with multiplicity $b-1$ and
the eigenvalues

$\dfrac{1}{2}\left(\alpha(a+b)\pm\sqrt{\alpha^{2}(a+b)^{2}+4ab(1-2\alpha)}\right).$

  * (ii)

The $A_{\alpha}$-spectrum of $CS_{\omega,n-\omega}$ is given by

$\left\\{(\alpha
n-1)^{(\omega-1)},(\alpha\omega)^{(n-\omega-1)},\frac{1}{2}\left(5n-2\omega-6\pm\sqrt{(3(2\omega-n)-2(\omega-1))^{2}+4\omega(n-\omega)}\right)\right\\}.$

  * (iii)

The $A_{\alpha}$-spectrum of $C_{a,b}$ consists of the eigenvalues
$\alpha(b+2)+2(1-\alpha)\cos\left(\frac{2\pi k}{b}\right)$, where
$k=2,\dots,b-1$, the eigenvalue $\alpha a$ with multiplicity $b-1$ and the
eigenvalues

$\dfrac{1}{2}\left(2+\alpha(a+b)\pm\sqrt{\alpha^{2}(a+b)^{2}+\alpha(4a-4b-8ab)+4ab+4}\right).$

  * (iv)

The $A_{\alpha}$-spectrum of $W_{n+1}$ consists of the eigenvalues
$3\alpha+2(1-\alpha)\cos\left(\frac{2\pi k}{n}\right)$, where $k=2,\dots,n-1$,
and the eigenvalues

$\dfrac{1}{2}\left(2+\alpha(n+1)\pm\sqrt{\alpha^{2}(n+1)^{2}+\alpha(4-12n)+4n+4}\right).$

Proof. (i). This follows from Corollary 2.10, by taking
$n_{1}=a,n_{2}=b,r_{1}=r_{2}=0$ and $\lambda_{1k}=0,$ for $k=2,\dots,a$ and
$\lambda_{2k}=0$ for each $k=2,\dots,b$.
(ii). We recall that the adjacency spectrum of $K_{\omega}$ is
$\\{\omega-1,-1^{(\omega-1)}\\}$. Now the result follows from Corollary 2.10
by taking
$n_{1}=\omega,n_{2}=n-\omega,r_{1}=\omega-1,r_{2}=0,\lambda_{1k}=-1,$ for
$k=2,\dots,\omega$ and $\lambda_{2k}=0$ for $k=2,3,\dots,n-\omega.$
(iii). We note that the adjacency spectrum (see [9]) of $C_{n}$ is
$\left\\{2\cos\left(\frac{2\pi k}{n}\right):k=1,2,\dots,n\right\\}$. Now
result follows from Corollary 2.10, by taking
$n_{1}=a,n_{2}=b,r_{1}=2,r_{2}=0$ and $\lambda_{1k}=2\cos\left(\frac{2\pi
i}{n}\right)$ for $k=2,3,\dots,n.$
(iv) Particular case of (iii) with $b=1$.

##  3 $A_{\alpha}$-Spectrum of power graphs of certain finite groups

Let $\mathcal{G}$ be a finite group of order $n$ with identity element $e$.
Chakrabarty et al. [7] defined the undirected power graph
$\mathcal{P}(\mathcal{G})$ of a group $\mathcal{G}$ as an undirected graph
with vertex set as $\mathcal{G}$ and two vertices $x,y\in\mathcal{G}$ are
adjacent if and only if one is the positive power of other, that is, $x^{i}=y$
or $y^{j}=x$, for positive integers $i,j$ with $2\leq i,j\leq n$. For some
recent work on power graphs, we refer to [5, 7, 8, 15, 1] and the references
therein. The adjacency spectrum, the Laplacian and the signless Laplacian
spectrum of power graphs of finite cyclic and dihedral groups has been
investigated in [3, 6, 11, 16, 20].

An integer $d$ dividing $n$ is written as $d|n$, and is called proper divisor
if $1<d<n.$ Let $d_{1},d_{2},\dots,d_{r}$ be the distinct proper divisors of
$n.$ Let $\mathbb{G}_{n}$ be a simple graph with vertex set
$\\{d_{i}:d_{i}|n,d_{i}\notin\\{1,n\\},~{}1\leq i\leq r\\}$ and edge set
$\\{d_{i}d_{j}:d_{i}|d_{j},~{}1\leq i<j\leq r\\}$. If the _canonical
decomposition_ of $n$ is $n=p_{1}^{n_{1}}p_{2}^{n_{2}}\dots p_{r}^{n_{r}}$,
where $r,n_{1},n_{2},\dots,n_{r}$ are positive integers and
$p_{1},p_{2},\dots,p_{r}$ are distinct prime numbers, then from elementary
number theory, the number of divisors of $n$ are
$\prod\limits_{i=1}^{r}(n_{i}+1)$, so the order of the graph $\mathbb{G}_{n}$
is $|V(\mathbb{G}_{n})|=\prod\limits_{i=1}^{r}(n_{i}+1)-2.$ If $n$ is neither
a prime power nor product of two distinct primes, then $\mathbb{G}_{n}$ is a
connected graph, see [23].

The following result shows that $\alpha n-1$ is always an
$A_{\alpha}$-eigenvalue of the power graph $\mathcal{P}(\mathcal{G})$.

###### Theorem 3.1

Let $\mathcal{G}$ be a finite group of order $n\geq 3$. Then $\alpha n-1$ is
always an $A_{\alpha}$-eigenvalue of the power graph
$\mathcal{P}(\mathcal{G})$ with multiplicity at least $b-1$, where $b$ is the
number of vertices which are adjacent to every other vertex of
$\mathcal{P}(\mathcal{G})$.

Proof. Let $B(\mathcal{G})$ be the subset of the vertex set $\mathcal{G}$ of
power graph $\mathcal{P}(\mathcal{G})$ consisting of the identity $e$ and
those vertices which are adjacent to every other vertex of
$\mathcal{P}(\mathcal{G})$. Assume that the cardinality of $B(\mathcal{G})$ is
$b$, so the induced subgraph of $\mathcal{G}$ induced by the vertices in
$B(\mathcal{G})$ is a complete graph $K_{b}$. Further, using the definition of
subset $B(\mathcal{G})$, we see that $\mathcal{P}(\mathcal{G})\cong
K_{b}\triangledown\mathcal{P}(\mathcal{G}\setminus B(\mathcal{G}))$. Applying
Corollary 2.8, we find that

$\alpha(r_{1}+n_{2})+(1-\alpha)\lambda_{1k}(G_{1})=\alpha(b-1+n-b)+(1-\alpha)(-1)=\alpha
n-1$

is an $A_{\alpha}$-eigenvalue of $\mathcal{P}(\mathcal{G})$ with multiplicity
$b-1$. Since $\alpha n-1$ can also be an eigenvalue of
$\mathcal{P}(\mathcal{G}\setminus B(\mathcal{G}))$, therefore it follows that
$\alpha n-1$ is an $A_{\alpha}$-eigenvalue of $\mathcal{P}(\mathcal{G})$ with
multiplicity at least $b-1$.

Taking in particular $\mathcal{G}=\mathbb{Z}_{n}$, a finite cyclic group of
order $n$, we have the following observation.

###### Corollary 3.2

Let $\mathbb{Z}_{n}$ be a finite cyclic group of order $n\geq 3$. Then $\alpha
n-1$ is an $A_{\alpha}$-eigenvalue of $\mathcal{P}(\mathbb{Z}_{n})$ with
multiplicity at least $\phi(n)$, where $\phi$ is Euler’s totient function.

Proof. Since identity $0$ and the invertible elements of the group
$\mathbb{Z}_{n}$ are adjacent to every other vertex of $\mathbb{Z}_{n}$ in the
power graph $\mathcal{P}(\mathbb{Z}_{n})$, the result follows by Theorem 3.1
and the fact that there are $\phi(n)$ invertible elements of $\mathbb{Z}_{n}$.

It will be an interesting problem to characterise the power graphs for which
equality holds in Theorem 3.1 and Corollary 3.2. Therefore, we leave the
following problem.

###### Problem 1

Characterize the power graphs $\mathcal{P}(\mathcal{G})$, where $\mathcal{G}$
is a finite group of order $n\geq 3$ having $\alpha n-1$ an
$A_{\alpha}$-eigenvalue with multiplicity $b-1$, where $b$ is the number of
vertices which are adjacent to every other vertex of
$\mathcal{P}(\mathcal{G})$. Also, characterize the power graphs
$\mathcal{P}(\mathbb{Z}_{n})$ having $\alpha n-1$ an $A_{\alpha}$-eigenvalue
with multiplicity $\phi(n)$.

From Theorem 3.1, we see that if $\mathcal{P}(\mathcal{G}\setminus
B(\mathcal{G})$ is known, then the $A_{\alpha}$-spectrum of
$\mathcal{P}(\mathcal{G})$ can be completely determined. So, it will be
interesting to study the structure of $\mathcal{P}(\mathcal{G}\setminus
B(\mathcal{G})$, and looking for graph parameters related to it.

The following result [15] shows that the power graph of a finite cyclic group
$\mathbb{Z}_{n}$ can be written as the joined union each of whose component is
a clique.

###### Theorem 3.3

If $\mathbb{Z}_{n}$ is a finite cyclic group of order $n\geq 3$, then the
power graph $\mathcal{P}(\mathbb{Z}_{n})$ is given by

$\mathcal{P}(\mathbb{Z}_{n})=K_{\phi(n)+1}\triangledown\mathbb{G}_{n}[K_{\phi(d_{1})},K_{\phi(d_{2})},\dots,K_{\phi(d_{r})}],$

where $\mathbb{G}_{n}$ is the graph of order $r$ defined above.

Using Theorems 2.3 and its corollaries, we can compute the
$A_{\alpha}$-spectrum of $\mathcal{P}(\mathbb{Z}_{n})$ in terms of the
adjacency spectrum of $K_{\omega}$ and zeros of the characteristic polynomial
of the auxiliary matrix. We recall [7] that $\mathcal{P}(\mathcal{G})$ is a
complete graph if and only if group $\mathcal{G}$ is a cyclic group of order
$n=p^{z}$, $z\in\mathbb{N}$, where $p$ is prime.

We form a new connected graph $H=K_{1}\triangledown\mathbb{G}_{n}$ which is of
diameter at most two if $\mathbb{G}_{n}$ is not complete, otherwise its
diameter is $1$. Now, we compute the $A_{\alpha}$-spectrum of the power graph
of $\mathbb{Z}_{n}$ by using Theorems 2.3 and Theorem 3.3.

###### Theorem 3.4

The $A_{\alpha}$-spectrum of $\mathcal{P}(\mathbb{Z}_{n})$ is

$\left\\{(\alpha
n-1)^{(\phi(n))},(\alpha(\phi(d_{1})+\alpha_{2})-1)^{(\phi(d_{1})-1)},\dots,(\alpha(\phi(d_{r})+\alpha_{r+1})-1)^{(\phi(d_{r})-1)}\right\\}$

and the eigenvalues of the matrix

$M=\begin{pmatrix}z_{1}&(1-\alpha)\phi(d_{1})&(1-\alpha)\phi(d_{2})&\dots&(1-\alpha)\phi(d_{n})\\\
(1-\alpha)(\phi(n)+1)&z_{2}&(1-\alpha)\psi_{23}&\dots&(1-\alpha)\psi_{2r}\\\
\vdots&\vdots&\vdots&\cdots&\vdots\\\
(1-\alpha)(\phi(n)+1)&(1-\alpha)\psi_{r2}&(1-\alpha)\psi_{r3}&\dots&z_{r+1}\end{pmatrix},$
(3.4)

where $z_{1}=\phi(n)+\alpha(n-1-\phi(n))$,
$z_{i}=\alpha\alpha_{i}+\phi(d_{i})-1,\alpha_{i}=\sum\limits_{v_{j}\in
N_{H}(v_{i})}n_{j}$, for $i=2,\dots,r+1$ and for $i\neq j$, $\psi_{ij}=n_{j}$,
if $v_{i}\sim v_{j}$, while as $\psi_{ij}=0$, if $v_{i}\nsim v_{j}$.

Proof. Let $\mathcal{G}\cong\mathbb{Z}_{n}$ be a finite cyclic group of order
$n$. Since by definition of power graph, the identity element $0$ and the
generators (which are $\phi(n)$ in number) of the group $\mathbb{Z}_{n}$, are
adjacent to every other vertex of $\mathcal{P}(\mathbb{Z}_{n})$. So, by
Theorem 3.3, we have

$\mathcal{P}(\mathbb{Z}_{n})=K_{\phi(n)+1}\triangledown\mathbb{G}_{n}[K_{\phi(d_{1})},K_{\phi(d_{2})},\dots,K_{\phi(d_{r})}]=H[K_{\phi(n)+1},K_{\phi(d_{1})},K_{\phi(d_{2})},\dots,K_{\phi(d_{r})}],$

where $H=K_{1}\triangledown\mathbb{G}_{n}$ is the graph with vertex set
$\\{v_{1},\dots,v_{r+1}\\}$. Taking $G_{1}=K_{\phi(n)+1}$ and
$G_{i}=K_{\phi(d_{i-1})}$, for $i=2,\dots,r+1$, in Theorem 2.3 and using the
fact that the adjacency spectrum of $K_{t}$ consists of the eigenvalue $t-1$
with multiplicity $1$ and the eigenvalue $-1$ with multiplicity $t-1$, it
follows that

$\displaystyle\alpha(r_{1}+\alpha_{1})+(1-\alpha)\lambda_{1k}A(G_{1})$
$\displaystyle=\alpha(\phi(n)+\phi(d_{1})+\dots+\phi(d_{r}))-1+\alpha$
$\displaystyle=\alpha(\phi(n)+n-1-\phi(n))-1+\alpha=\alpha n-1$

is an $A_{\alpha}$ eigenvalues of $\mathcal{P}(\mathbb{Z}_{n})$ with
multiplicity $\phi(n)$. Note that we have used the fact that vertex $v_{1}$ of
graph $H$ is adjacent to every other vertex of $H$ and
$\alpha_{1}=\sum\limits_{d|n,d\neq 1,n}\phi(d)=n-1-\phi(n)$, as
$\sum\limits_{d|s}\phi(d)=s.$ Similarly, we can show that
$(\alpha(\phi(d_{1})+\alpha_{2})-1),\dots,(\alpha(\phi(d_{r})+\alpha_{r+1})-1)$
are the $A_{\alpha}$-eigenvalues of $\mathcal{P}(\mathbb{Z}_{n})$ with
multiplicities $\phi(d_{1})-1,\dots,\phi(d_{r})-1$ respectively. The remaining
$A_{\alpha}$-eigenvalues are the zeros of the characteristic polynomial of the
equitable quotient matrix $M$ given by (3.4).

Clearly, from Theorem 3.4 all the $A_{\alpha}$-eigenvalues of the power graph
$\mathcal{P}(\mathbb{Z}_{n})$ are completely determined except the
$r+1$-eigenvalues, which are the eigenvalues of the matrix $M$ defined in
Theorem 3.4. Further, it is also clear that the matrix $M$ depends upon the
structure of the graph $\mathbb{G}_{n}$, which is not known in general.
However, if we give some particular values to $n$, then it may be possible to
know the structure of graph $\mathbb{G}_{n}$ and hence about the matrix $M$.
This information can be helpful to determine the $r+1$ remaining
$A_{\alpha}$-eigenvalues of the power graph $\mathcal{P}(\mathbb{Z}_{n})$.

We have following consequences of Theorem 3.4.

###### Corollary 3.5

If $n=p^{z}$, where $p$ is prime and $z$ is a positive integer, then the
$A_{\alpha}$-spectrum of $\mathcal{P}(\mathbb{Z}_{n})$ is $\\{n-1,(\alpha
n-1)^{(n-1)}\\}$.

Proof. If $n=p^{z}$, where $p$ is prime and $z\in\mathbb{N}$, then as shown in
[7], $\mathcal{P}(\mathbb{Z}_{n})$ is isomorphic to the complete graph $K_{n}$
and hence the result follows.

###### Corollary 3.6

If $n=pq$, where $p$ and $q$ ($p<q$) are primes, then the
$A_{\alpha}$-spectrum of $\mathcal{P}(\mathbb{Z}_{n})$ consists of eigenvalues
$\\{(\alpha
n-1)^{(\phi(n))},(\alpha(\phi(n)+\phi(p)+1)-1)^{(\phi(p)-1)},(\alpha(\phi(n)+\phi(q)+1)-1)^{(\phi(q)-1)}\\}$
and the three eigenvalues of the matrix given by (3.5).

Proof. Let $n=pq$, where $p$ and $q$, $p<q$, are distinct primes. Clearly,
there are $\phi(pq)$ generators in $\mathbb{Z}_{pq}$. These generators and
identity element are adjacent to every other vertex of
$\mathcal{P}(\mathbb{Z}_{n})$. So, by [8], we have

$\mathcal{P}(\mathbb{Z}_{n})=\left(K_{\phi(p)}\cup
K_{\phi(q)}\right)\triangledown
K_{\phi(pq)+1}=P_{3}[K_{\phi(p)},K_{\phi(pq)+1},K_{\phi(q)}].$

Now, letting $G_{1}=K_{\phi(pq)+1},G_{2}=K_{\phi(p)}$ and $G_{3}=K_{\phi(q)}$
in Corollary 2.7, we obtain the required $A_{\alpha}$-eigenvalues and the
remaining three eigenvalues are given by the matrix

$\begin{pmatrix}\phi(p)-1+\alpha(\phi(pq)+1)&(1-\alpha)(\phi(pq)+1)&0\\\
(1-\alpha)\phi(p)&\phi(pq)+\alpha(\phi(p)+\phi(q))&(1-\alpha)\phi(q)\\\
0&(1-\alpha)(\phi(pq)+1)&\phi(q)-1+\alpha(\phi(pq)+1)\end{pmatrix}.$ (3.5)

Now, let $n=pqr$, where $p,~{}q,~{}r$ with $p<q<r$ are primes. From the
definition of $\mathbb{G}_{n}$, the vertex set and edge set of
$\mathbb{G}_{n}$ are $\\{p,q,r,pq,pr,qr\\}$ and
$\\{(p,pq),(p,pr),(q,pq),(q,qr),(r,pr),(r,qr)\\}$. Let
$H=K_{1}\triangledown\mathbb{G}_{n}.$ Then
$\mathcal{P}(\mathbb{Z}_{n})=H[K_{\phi(n)+1},K_{\phi(d_{1})},\dots,K_{\phi(d_{6})}]$.

By Corollary 3.2, $\alpha n-1$ is the $A_{\alpha}$-eigenvalue with
multiplicity $\phi(n)$. By Theorem 2.3, other $A_{\alpha}$-eigenvalues are
$\alpha(1+\phi(n)+\phi(p)+\phi(pr)+\phi(pq))-1,~{}\alpha(1+\phi(n)+\phi(q)+\phi(qr)+\phi(pq))-1,~{}\alpha(1+\phi(n)+\phi(r)+\phi(pr)+\phi(qr))-1,~{}\alpha(1+\phi(n)+\phi(pq)+\phi(p)+\phi(q))-1,~{}\alpha(1+\phi(n)+\phi(pr)+\phi(p)+\phi(r))-1,~{}\alpha(1+\phi(n)+\phi(qr)+\phi(q)+\phi(r))-1$
with multiplicities
$\phi(p)-1,~{}\phi(q)-1,~{}\phi(r)-1,~{}\phi(pq)-1,~{}\phi(pr)-1,~{}\phi(qr)-1$,
respectively. The remaining $7$ eigenvalues are given by the following matrix

$\begin{pmatrix}z_{1}&(1-\alpha)\phi(p)&(1-\alpha)\phi(q)&(1-\alpha)\phi(r)&(1-\alpha)\phi(pq)&(1-\alpha)\phi(pr)&(1-\alpha)\phi(qr)\\\
c_{1}&z_{2}&0&0&(1-\alpha)\phi(pq)&(1-\alpha)\phi(pr)&0\\\
c_{1}&0&z_{3}&0&(1-\alpha)\phi(pq)&0&(1-\alpha)\phi(qr)\\\
c_{1}&0&0&z_{4}&0&(1-\alpha)\phi(pr)&(1-\alpha)\phi(qr)\\\
c_{1}&(1-\alpha)\phi(p)&(1-\alpha)\phi(q)&0&z_{5}&0&0\\\
c_{1}&(1-\alpha)\phi(p)&0&(1-\alpha)\phi(r)&0&z_{6}&0\\\
c_{1}&0&(1-\alpha)\phi(q)&(1-\alpha)\phi(r)&0&0&z_{7}\end{pmatrix},$ (3.6)

where,
$z_{1}=\phi(n)+\alpha(n-\phi(n)-1),~{}z_{2}=\phi(p)-1+\alpha(\phi(n)+1+\phi(pr)+\phi(pq)),~{}z_{3}=\phi(q)-1+\alpha(\phi(n)+1+\phi(pq)+\phi(pr)),~{}z_{4}=\phi(r)-1+\alpha(\phi(n)+1+\phi(pr)+\phi(qr)),~{}z_{5}=\phi(pq)-1+\alpha(\phi(n)+1+\phi(p)+\phi(q)),~{}z_{6}=\phi(pr)-1+\alpha(\phi(n)+1+\phi(p)+\phi(r)),~{}z_{7}=\phi(qr)-1+\alpha(\phi(n)+1+\phi(q)+\phi(r))$
and $c_{1}=(1-\alpha)(\phi(n)+1).$

For the finite cyclic group $\mathbb{Z}_{pq^{N}}$, where $p$ and $q$ are
distinct primes and $N$ is a positive integer, we have the following
observation.

###### Theorem 3.7

Let $\mathcal{P}(\mathbb{Z}_{pq^{N}})$ be the power graph of the finite cyclic
group $\mathbb{Z}_{pq^{N}}$ of order $n=pq^{N}$, where $p$ and $q$ are
distinct primes and $N$ a positive integer. Then the following hold.

  * (i)

If $N$ is even, say $N=2m$. Then $A_{\alpha}$-spectrum of
$\mathcal{P}(\mathbb{Z}_{pq^{N}})$ is

$\displaystyle\\{$ $\displaystyle(\alpha
n-1)^{(\phi(n))},\left(\alpha(\phi(n)+\phi(p)q^{2m-1}+1)-1\right)^{(\phi(p)-1)},(\alpha(\phi(n)+q^{2m}+\phi(p)(q^{2m-1}-1))$
$\displaystyle-1)^{(\phi(q)-1)},\dots,\left(\alpha(\phi(n)+q^{2m}+\phi(p)(q^{2m-1}-q^{m-1}))-1\right)^{(\phi(q^{m})-1)},\dots,(\alpha(\phi(n)+q^{2m})$
$\displaystyle-1)^{(\phi(q^{2m})-1)},\left(\alpha(\phi(n)+\phi(q)+\phi(p)q^{2m-1}+1)-1\right)^{(\phi(pq)-1)},\dots,(\alpha(\phi(n)+q^{m}+\phi(p)q^{2m-1})$
$\displaystyle-1)^{(\phi(pq^{m})-1)},\dots,\left(\alpha(\phi(n)+q^{2m-1}+\phi(p)q^{2m-1})-1\right)^{(\phi(pq^{2m-1})-1)}\\}$

together with the zeros of the characteristic polynomial of the matrix given
by (3.7).

  * (ii)

If $N$ is odd, say $N=2m+1$. Then $A_{\alpha}$-spectrum of
$\mathcal{P}(\mathbb{Z}_{pq^{N}})$ is

$\displaystyle\\{(\alpha n$
$\displaystyle-1)^{(\phi(n))},\left(\alpha(\phi(n)+\phi(p)q^{2m}+1)-1\right)^{(\phi(p)-1)},\left(\alpha(\phi(n)+q^{2m+1}+\phi(p)(q^{2m}-1))-1\right)^{(\phi(q)-1)},$
$\displaystyle\dots,$
$\displaystyle\left(\alpha(\phi(n)+q^{2m+1}+\phi(p)(q^{2m}-q^{m}))-1\right)^{(\phi(q^{m+1})-1)},\dots,\left(\alpha(\phi(n)+q^{2m+1})-1\right)^{(\phi(q^{2m+1})-1)},$
$\displaystyle\left(\alpha(\phi(n)+\phi(q)+\phi(p)q^{2m}+1)-1\right)^{(\phi(pq)-1)},\dots,\left(\alpha(\phi(n)+q^{m+1}+\phi(p)q^{2m})-1\right)^{(\phi(pq^{m+1})-1)},$
$\displaystyle\dots,$
$\displaystyle\left(\alpha(\phi(n)+q^{2m}+\phi(p)q^{2m-1})-1\right)^{(\phi(pq^{2m}-1)}\\}$

together with the zeros of the characteristic polynomial of the matrix given
by (3.8).

Proof. (i). Let $N$ be even, say $N=2m,m\in\mathbb{N}$. Clearly, the order of
$\mathbb{G}_{pq^{2m}}$ is $n=4m$ and its vertex set is
$\\{p,q,q^{2},\dots,q^{m},\dots,q^{2m},pq,pq^{2},\dots,pq^{m},\dots,pq^{2m-1}\\}$,
where $p$ and $q$ are distinct primes. Also, in the graph
$\mathbb{G}_{pq^{N}}$, we have $p\sim pq^{i}$, for $i=1,2,\dots,2m-1$,
$q^{i}\sim q^{j},$ for all $1\leq i<j\leq 2m$, $q^{i}\sim pq^{j}$, for all
$1\leq i\leq j\leq 2m-1$ and $pq^{i}\sim pq^{j}$ for all $1\leq i<j\leq 2m-1$,
where we are avoiding the divisor which are outside the vertex set of
$\mathbb{G}_{pq^{2m}}$. Thus, by Theorem 3.3, we have

$\displaystyle\mathcal{P}(\mathbb{Z}_{pq^{N}})$
$\displaystyle=K_{\phi(pq^{N})+1}\triangledown\mathbb{G}_{pq^{N}}[K_{\phi(p)},K_{\phi(q)},\dots,K_{\phi(q^{m})},\dots,K_{\phi(q^{2m})},K_{\phi(pq)},\dots,K_{\phi(pq^{m})},\dots,K_{\phi(pq^{2m-1})}]$
$\displaystyle=H[K_{\phi(pq^{N})+1},K_{\phi(p)},K_{\phi(q)},\dots,K_{\phi(q^{m})},\dots,K_{\phi(q^{2m})},K_{\phi(pq)},\dots,K_{\phi(pq^{m})},\dots,K_{\phi(pq^{2m-1})}],$

where $H=K_{1}\triangledown\mathbb{G}_{pq^{N}}$. By Corollary 3.2, we find
that $\alpha n-1$ is an $A_{\alpha}$-eigenvalue with multiplicity $\phi(n)$.
Now, by using the Theorem 2.3 with
$n_{1}=\phi(pq^{2m})+1,~{}n_{2}=\phi(p),~{}n_{i}=\phi(q^{j}),i=3,\dots,2m+2,~{}j=1,2,\dots,2m,~{}n_{i}=\phi(pq^{j}),i=2m+3,\dots,4m+1,~{}j=1,2,\dots,2m-1$,
it follows that

$\displaystyle\alpha(\phi(p)$
$\displaystyle-1+\phi(n)+1+\phi(pq)+\dots+\phi(pq^{m})+\dots+\phi(pq^{2m-1}))+(1-\alpha)(-1)$
$\displaystyle=\alpha(\phi(n)+\phi(p)(1+\phi(q)+\dots+\phi(q^{2m-1})))-1+\alpha=\alpha(\phi(n)+\phi(p)q^{2m-1}+1)-1$

is the $A_{\alpha}$-eigenvalue with multiplicity $\phi(p)-1$. Note that we
have used the fact that $\phi(xy)=\phi(x)\phi(y),$ provided that g.c.d. of
$x,y$ is $1$ and $\sum\limits_{i=1}^{k}\phi(p^{i})=p^{k}-1$. Using the similar
procedure we get the other $A_{\alpha}$-eigenvalues as given in the statement.
The remaining $A_{\alpha}$-eigenvalues are given by the quotient matrix

$\left({\begin{array}[]{c c c c c c c c c c c c c
c}z_{1}&n_{2}^{{}^{\prime}}&n_{3}^{{}^{\prime}}&\cdots&n_{m+2}^{{}^{\prime}}&n_{m+3}^{{}^{\prime}}&\cdots&n_{2m+2}^{{}^{\prime}}&n_{2m+3}^{{}^{\prime}}&\cdots&n_{3m+2}^{{}^{\prime}}&n_{3m+3}^{{}^{\prime}}&\cdots&n_{4m+1}^{{}^{\prime}}\\\
n_{1}^{{}^{\prime}}&z_{2}&0&\cdots&0&0&\cdots&0&n_{2m+3}^{{}^{\prime}}&\cdots&n_{3m+2}^{{}^{\prime}}&n_{3m+3}^{{}^{\prime}}&\cdots&n_{4m+1}^{{}^{\prime}}\\\
n_{1}^{{}^{\prime}}&0&z_{3}&\cdots&n_{m+2}^{{}^{\prime}}&n_{m+3}^{{}^{\prime}}&\cdots&n_{2m+2}^{{}^{\prime}}&n_{2m+3}^{{}^{\prime}}&\cdots&n_{3m+2}^{{}^{\prime}}&n_{3m+3}^{{}^{\prime}}&\cdots&n_{4m+1}^{{}^{\prime}}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
n_{1}^{{}^{\prime}}&0&n_{3}^{{}^{\prime}}&\cdots&z_{m+2}&n_{m+3}^{{}^{\prime}}&\cdots&n_{2m+2}^{{}^{\prime}}&0&\cdots&n_{3m+2}^{{}^{\prime}}&n_{3m+3}^{{}^{\prime}}&\cdots&n_{4m+1}^{{}^{\prime}}\\\
n_{1}^{{}^{\prime}}&n_{2}^{{}^{\prime}}&n_{3}^{{}^{\prime}}&\cdots&n_{m+2}^{{}^{\prime}}&z_{m+3}&\cdots&n_{2m+2}^{{}^{\prime}}&0&\cdots&0&0&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
n_{1}^{{}^{\prime}}&n_{2}^{{}^{\prime}}&n_{3}^{{}^{\prime}}&\cdots&n_{m+2}^{{}^{\prime}}&n_{m+3}^{{}^{\prime}}&\cdots&z_{2m+2}^{{}^{\prime}}&0&\cdots&0&0&\cdots&0\\\
n_{1}^{{}^{\prime}}&n_{2}^{{}^{\prime}}&n_{3}^{{}^{\prime}}&\cdots&n_{m+2}^{{}^{\prime}}&n_{m+3}^{{}^{\prime}}&\cdots&n_{2m+2}^{{}^{\prime}}&z_{2m+3}^{{}^{\prime}}&\cdots&n_{3m+2}^{{}^{\prime}}&n_{3m+3}^{{}^{\prime}}&\cdots&n_{4m+1}^{{}^{\prime}}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
n_{1}^{{}^{\prime}}&n_{2}^{{}^{\prime}}&n_{3}^{{}^{\prime}}&\cdots&0&0&\cdots&0&n_{2m+3}^{{}^{\prime}}&\cdots&z_{3m+2}^{{}^{\prime}}&n_{3m+3}^{{}^{\prime}}&\cdots&n_{4m+1}^{{}^{\prime}}\\\
n_{1}^{{}^{\prime}}&n_{2}^{{}^{\prime}}&n_{3}^{{}^{\prime}}&\cdots&n_{m+2}^{{}^{\prime}}&n_{m+3}^{{}^{\prime}}&\cdots&0&n_{2m+3}^{{}^{\prime}}&\cdots&n_{3m+2}^{{}^{\prime}}&z_{3m+3}^{{}^{\prime}}&\cdots&n_{4m+1}^{{}^{\prime}}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
n_{1}^{{}^{\prime}}&n_{2}^{{}^{\prime}}&n_{3}^{{}^{\prime}}&\cdots&n_{m+2}^{{}^{\prime}}&n_{m+3}^{{}^{\prime}}&\cdots&0&n_{2m+3}^{{}^{\prime}}&\cdots&n_{3m+2}^{{}^{\prime}}&n_{3m+3}^{{}^{\prime}}&\cdots&z_{4m+1}^{{}^{\prime}}\\\
\end{array}}\right),$ (3.7)

where $n_{i}^{{}^{\prime}}=(1-\alpha)n_{i},~{}\text{for}~{}i=1,2,\dots,4m+1$
and
$z_{1}=\phi(n)+\alpha(n-\phi(n)-1),~{}z_{2}=\phi(p)-1+\alpha(\phi(n)+1+\phi(p)(q^{2m-1}-q)),~{}z_{3}=\phi(q)-1+\alpha(\phi(n)+1+q^{2m}-q+\phi(p)(q^{2m-1}-1)),\dots,z_{m+2}=\phi(q^{m})-1+\alpha(\phi(n)+q^{2m}-\phi(q^{m})+\phi(p)(q^{2m}-q^{m-1})),z_{m+3}=\phi(q^{m+1})-1+\alpha(\phi(n)+q^{2m}-\phi(q^{m+1})+\phi(p)(q^{2m}-q^{m})),\dots,z_{2m+2}=\phi(q^{2m})-1+\alpha(\phi(n)+q^{2m}-\phi(q^{2m})),z_{2m+3}=\phi(pq)-1+\alpha(\phi(n)+\phi(p)+\phi(q)+\phi(p)(q^{2m-1}-q)),\dots,z_{3m+2}=\phi(pq^{m})-1+\alpha(\phi(n)+\phi(p)+q^{m}+\phi(p)(q^{2m-1}-1-\phi(q^{m}))),z_{3m+3}=\phi(pq^{m+1})-1+\alpha(\phi(n)+\phi(p)+q^{m+1}+\phi(p)(q^{2m}-1-\phi(q^{m+1}))),\dots,z_{4m+1}=\phi(pq^{2m-1})-1+\alpha(\phi(n)+q^{2m-1}+\phi(p)q^{2m-2}).$

(ii). Now, let $N$ be odd, say $N=2m+1,m\in\mathbb{N}$. Clearly, the order of
$\mathbb{G}_{pq^{2m+1}}$ is $4m+2$ and its vertex set is
$\\{p,q,q^{2},\dots,q^{m+1},\dots,q^{2m+1},pq,pq^{2},\dots,pq^{m+1},\dots,pq^{2m}\\}$,
where $p$ and $q$ are distinct primes. Also, in $\mathbb{G}_{pq^{2m+1}}$, we
have $p\sim pq^{i}$, for $i=1,2,\dots,N-1$, $q^{i}\sim q^{j}$, for all $1\leq
i<j\leq 2m+1$, $q^{i}\sim pq^{j}$, for all $1\leq i\leq j\leq 2m$ and
$pq^{i}\sim pq^{j},$ for all $1\leq i<j\leq 2m$. Note that we are avoiding the
divisor which are outside the vertex set of $\mathbb{G}_{pq^{2m+1}}$.
Therefore, by Theorem 3.3, we have

$\displaystyle\mathcal{P}(\mathbb{Z}_{pq^{N}})=K_{\phi(pq^{N})+1}$
$\displaystyle\triangledown\mathbb{G}_{pq^{N}}[K_{\phi(p)},K_{\phi(q)},K_{\phi(q^{2})},\dots,K_{\phi(q^{m+1})},\dots,K_{\phi(q^{N})},K_{\phi(pq)},\dots,K_{\phi(pq^{m+1})},\dots,$
$\displaystyle
K_{\phi(pq^{N-1})}]=H[K_{\phi(pq^{N})+1},K_{\phi(p)},K_{\phi(q)},K_{\phi(q^{2})},\dots,K_{\phi(q^{m+1})},\dots,K_{\phi(q^{N})},K_{\phi(pq)},$
$\displaystyle\dots,K_{\phi(pq^{m+1})},\dots,K_{\phi(pq^{N-1})}],$

where $H=K_{1}\triangledown\mathbb{G}_{pq^{N}}$. By Corollary 3.2, it follows
that $\alpha n-1$ is an $A_{\alpha}$-eigenvalue with multiplicity $\phi(n)$.
Using Theorem 2.3 with
$n_{1}=\phi(pq^{2m+1})+1,~{}n_{2}=\phi(p),~{}n_{i}=\phi(q^{j}),i=2,\dots,2m+3,~{}j=1,2,\dots,2m+1,~{}n_{i}=\phi(pq^{j}),i=2m+3,\dots,4m+2,~{}j=1,2,\dots,2m$,
it follows that $\alpha(\phi(n)+\phi(p)+\phi(p)(q^{2m}-1)+1)-1$ is the
$A_{\alpha}$-eigenvalue with multiplicity $\phi(p)-1$. Proceeding in a similar
way, we get the other $A_{\alpha}$-eigenvalues as given in statement. The
remaining $A_{\alpha}$-eigenvalues are given by the following quotient matrix

$\left({\begin{array}[]{c c c c c c c c c c c c c
c}z_{1}&n_{2}^{{}^{\prime}}&n_{3}^{{}^{\prime}}&\cdots&n_{m+3}^{{}^{\prime}}&n_{m+4}^{{}^{\prime}}&\cdots&n_{2m+3}^{{}^{\prime}}&n_{2m+4}^{{}^{\prime}}&\cdots&n_{3m+3}^{{}^{\prime}}&n_{3m+4}^{{}^{\prime}}&\cdots&n_{4m+2}^{{}^{\prime}}\\\
n_{1}^{{}^{\prime}}&z_{2}&0&\cdots&0&0&\cdots&0&n_{2m+4}^{{}^{\prime}}&\cdots&n_{3m+3}^{{}^{\prime}}&n_{3m+4}^{{}^{\prime}}&\cdots&n_{4m+2}^{{}^{\prime}}\\\
n_{1}^{{}^{\prime}}&0&z_{3}&\cdots&n_{m+3}^{{}^{\prime}}&n_{m+4}^{{}^{\prime}}&\cdots&n_{2m+3}^{{}^{\prime}}&n_{2m+4}^{{}^{\prime}}&\cdots&n_{3m+3}^{{}^{\prime}}&n_{3m+4}^{{}^{\prime}}&\cdots&n_{4m+2}^{{}^{\prime}}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
n_{1}^{{}^{\prime}}&0&n_{3}^{{}^{\prime}}&\cdots&z_{m+3}&n_{m+4}^{{}^{\prime}}&\cdots&n_{2m+3}^{{}^{\prime}}&0&\cdots&n_{3m+3}^{{}^{\prime}}&n_{3m+4}^{{}^{\prime}}&\cdots&n_{4m+2}^{{}^{\prime}}\\\
n_{1}^{{}^{\prime}}&n_{2}^{{}^{\prime}}&n_{3}^{{}^{\prime}}&\cdots&n_{m+3}^{{}^{\prime}}&z_{m+4}&\cdots&n_{2m+3}^{{}^{\prime}}&0&\cdots&0&0&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
n_{1}^{{}^{\prime}}&n_{2}^{{}^{\prime}}&n_{3}^{{}^{\prime}}&\cdots&n_{m+3}^{{}^{\prime}}&n_{m+4}^{{}^{\prime}}&\cdots&z_{2m+3}^{{}^{\prime}}&0&\cdots&0&0&\cdots&0\\\
n_{1}^{{}^{\prime}}&n_{2}^{{}^{\prime}}&n_{3}^{{}^{\prime}}&\cdots&n_{m+3}^{{}^{\prime}}&n_{m+4}^{{}^{\prime}}&\cdots&n_{2m+3}^{{}^{\prime}}&z_{2m+4}^{{}^{\prime}}&\cdots&n_{3m+3}^{{}^{\prime}}&n_{3m+4}^{{}^{\prime}}&\cdots&n_{4m+2}^{{}^{\prime}}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
n_{1}^{{}^{\prime}}&n_{2}^{{}^{\prime}}&n_{3}^{{}^{\prime}}&\cdots&0&0&\cdots&0&n_{2m+4}^{{}^{\prime}}&\cdots&z_{3m+3}^{{}^{\prime}}&n_{3m+4}^{{}^{\prime}}&\cdots&n_{4m+2}^{{}^{\prime}}\\\
n_{1}^{{}^{\prime}}&n_{2}^{{}^{\prime}}&n_{3}^{{}^{\prime}}&\cdots&n_{m+3}^{{}^{\prime}}&n_{m+4}^{{}^{\prime}}&\cdots&0&n_{2m+4}^{{}^{\prime}}&\cdots&n_{3m+3}^{{}^{\prime}}&z_{3m+4}^{{}^{\prime}}&\cdots&n_{4m+2}^{{}^{\prime}}\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots\\\
n_{1}^{{}^{\prime}}&n_{2}^{{}^{\prime}}&n_{3}^{{}^{\prime}}&\cdots&n_{m+3}^{{}^{\prime}}&n_{m+4}^{{}^{\prime}}&\cdots&0&n_{2m+4}^{{}^{\prime}}&\cdots&n_{3m+3}^{{}^{\prime}}&n_{3m+4}^{{}^{\prime}}&\cdots&z_{4m+2}^{{}^{\prime}}\\\
\end{array}}\right),$ (3.8)

where $n_{i}^{{}^{\prime}}=(1-\alpha)n_{i},~{}\text{for}~{}i=1,2,\dots,4m+1$
and
$z_{1}=\phi(n)+\alpha(n-\phi(n)-1),~{}z_{2}=\phi(p)-1+\alpha(\phi(n)+1+\phi(p)(q^{2m}-1)),~{}z_{3}=\phi(q)-1+\alpha(\phi(n)+1+q^{2m+1}-q+\phi(p)(q^{2m}-1)),\dots,z_{m+3}=\phi(q^{m+1})-1+\alpha(\phi(n)+q^{2m+1}-\phi(q^{m+1})+\phi(p)(q^{2m}-q^{m})),z_{m+4}=\phi(q^{m+2})-1+\alpha(\phi(n)+q^{2m+1}-\phi(q^{m+2})+\phi(p)(q^{2m}-q^{m+1})),\dots,z_{2m+3}=\phi(q^{2m+1})-1+\alpha(\phi(n)+q^{2m+1}),z_{2m+4}=\phi(pq)-1+\alpha(\phi(n)+\phi(p)+\phi(q)+\phi(p)(q^{2m}-q)),\dots,z_{3m+3}=\phi(pq^{m+1})-1+\alpha(\phi(n)+\phi(p)+q^{m+1}+\phi(p)(q^{2m}-1-\phi(q^{m+1}))),z_{3m+4}=\phi(pq^{m+2})-1+\alpha(\phi(n)+\phi(p)+q^{m+2}+\phi(p)(q^{2m}-1-\phi(q^{m+2}))),\dots,z_{4m+2}=\phi(pq^{2m})-1+\alpha(\phi(n)+q^{2m}+\phi(p)q^{2m-1}).$
This completes the proof.

Although for $n=p^{z},pq,pqr,pq^{N}$, where $p,q,r$ are distinct primes and
$z,N$ are positive integers, we have described the $A_{\alpha}$-spectrum of
the power graph $\mathcal{P}(\mathbb{Z}_{n})$, the following problem will be
of interest.

###### Problem 2

Find the $A_{\alpha}$-spectrum of the power graph
$\mathcal{P}(\mathbb{Z}_{n})$ of the finite cyclic group $\mathbb{Z}_{n}$,
when $n\in\\{(pq)^{2},~{}pqrs,~{}(pq)^{m},~{}m\in\mathbb{N}\\}$ and generalize
(if possible) for any $n$.

A group $\mathcal{G}$ is an elementary abelian $p$-group, if each non trivial
element has order $p$, where $p$ is prime. In order to find
$A_{\alpha}$-spectrum of finite elementary abelian groups of prime power
order, we need following lemma [8].

###### Lemma 3.8

Let $\mathcal{G}$ be an elementary abelian group of order $p^{n}$ for some
prime number $p$ and positive integer $n.$ Then $\mathcal{P}(\mathcal{G})\cong
K_{1}\triangledown\left(\cup_{i=1}^{l}K_{p-1}\right),$ where
$l=\frac{p^{n}-1}{p-1}.$

The following result gives the $A_{\alpha}$-spectrum of power graph of
elementary $p$-group of order $p^{n}$.

###### Theorem 3.9

Let $\mathcal{G}$ be an elementary abelian $p$-group of order $p^{n}$. The
$A_{\alpha}$-spectrum of $\mathcal{P}(\mathcal{G})$ consists of the eigenvalue
$\alpha p-1$ with multiplicity $l(p-2)$, where $l=\frac{p^{n}-1}{p-1}$ and the
eigenvalues of the matrix given by (3.9).

Proof. Let $\mathcal{G}$ be an elementary abelian $p$-group of order $p^{n}$
for some prime $p$ and positive integer $n.$ Then, by Lemma 3.8, we have
$\mathcal{P}(\mathcal{G})\cong
K_{1}\triangledown\left(\cup_{i=1}^{l}K_{p-1}\right)$ and so it follows that

$\mathcal{P}(\mathcal{G})\cong
S[K_{1},\underbrace{K_{p-1},\dots,K_{p-1}}_{l}],$

where $K_{p-1}$ occurs $l=\frac{p^{n}-1}{p-1}$ times and
$S=K_{1}\triangledown\overline{K}_{l}=K_{1,l}.$ Since
$G_{1}=K_{1},~{}G_{i}=K_{p-1}$, for $i=2,\dots,l+1$ and
$\alpha_{1}=l(p-1)=p^{n}-1$, $\alpha_{2}=\dots=\alpha_{l+1}=1$, therefore,
using Theorem 2.3, we see that
$\alpha(r_{2}+\alpha_{2})+(1-\alpha)\lambda_{2k}A(G_{2})=\alpha(p-2+1)+\alpha-1=\alpha
p-1$ is the $A_{\alpha}$-eigenvalue of $\mathcal{P}(\mathcal{G})$ with
multiplicity $l(p-2)$. The $l+1$ remaining $A_{\alpha}$-eigenvalues of
$\mathcal{P}(\mathcal{G})$ are given by the matrix

$\begin{pmatrix}\alpha
l(p-1)&(1-\alpha)(p-1)&(1-\alpha)(p-1)&\dots&(1-\alpha)(p-1)\\\
1-\alpha&\alpha+p-2&0&\dots&0\\\ \vdots&\vdots&\vdots&\cdots&\vdots\\\
1-\alpha&0&0&\cdots&\alpha+p-2\end{pmatrix}.$ (3.9)

It is easy to see that $\alpha+p-2$ with multiplicity $l-1$ is the eigenvalue
of the matrix given by (3.9) and the other two eigenvalues are given by the
following matrix

$\begin{pmatrix}\alpha l(p-1)&l(1-\alpha)(p-1)\\\
1-\alpha&\alpha+p-2\end{pmatrix},$

which are calculated as $\alpha(lp+1-l)+p-2\pm\sqrt{(\alpha lp+p+\alpha-\alpha
l-2)^{2}-4l(\alpha p^{2}-\alpha p-p+1)}$. This completes the proof.

The following observation can be seen in [8].

###### Lemma 3.10

Let $\mathcal{G}$ be a finite group of order $pq$, where $p$ and $q$ are
primes. Then $\mathcal{G}$ is non abelian if and only if
$\mathcal{P}(\mathcal{G})\cong K_{1}\triangledown(qK_{p-1}\cup K_{q-1})$.

Now, we obtain the $A_{\alpha}$-spectrum of the non abelian group whose order
is the product of two distinct primes.

###### Theorem 3.11

Let $\mathcal{G}$ be a non abelian group of order $n=pq$, where $p,~{}q$
($p<q$) are primes. Then the $A_{\alpha}$-spectrum of
$\mathcal{P}(\mathcal{G})$ consists of the eigenvalue $\alpha p-1$ with
multiplicity $q(p-2)$, the eigenvalue $\alpha q-1$ with multiplicity $q-2$ and
the eigenvalues of the matrix given by (3.10).

Proof. Let $\mathcal{G}$ be a non abelian group of order $pq$. By Lemma 3.10,
the power graph of $\mathcal{P}(\mathcal{G})$ can be written as

$\mathcal{P}(\mathcal{G})=K_{1,q+1}[K_{1},\underbrace{K_{p-1},K_{p-1},\dots,K_{p-1}}_{q},K_{q-1}].$

Since $\alpha_{1}=pq-1,~{}\alpha_{2}=\dots=\alpha_{q+2}=1$, so by Theorem 2.3,
$\alpha p-1$ is the $A_{\alpha}$-eigenvalue with multiplicity $q(p-1)$.
Similarly $\alpha q-1$ is the $A_{\alpha}$-eigenvalue with multiplicity $q-2$
and the remaining $q+2$ $A_{\alpha}$-eigenvalues of $\mathcal{P}(\mathcal{G})$
are the eigenvalues of the matrix $M$ given in (3.10).

$M=\begin{pmatrix}\alpha(pq-1)&(1-\alpha)(p-1)&(1-\alpha)(p-1)&\dots&(1-\alpha)(p-1)&(1-\alpha)(q-1)\\\
1-\alpha&\alpha+p-2&0&\dots&0&0\\\
\vdots&\vdots&\vdots&\cdots&\vdots&\vdots\\\
1-\alpha&0&0&\cdots&\alpha+p-2&0\\\ 1-\alpha&0&0&\cdots&0&\alpha+q-2\\\
\end{pmatrix}$ (3.10)

Clearly, $\alpha+p-2$ is an $A_{\alpha}$-eigenvalue of matrix $M$ with
multiplicity $q-1$. The remaining three eigenvalues of matrix $M$ are the
eigenvalues of the following matrix

$\begin{pmatrix}\alpha l(p-1)&q(1-\alpha)(p-1)&(1-\alpha)(q-1)\\\
1-\alpha&\alpha+p-2&0\\\ 1-\alpha&0&\alpha+q-2\end{pmatrix}.$

Next, we find the $A_{\alpha}$-spectrum of the dihedral group and dicyclic
group for some particular values of $n$. The dihedral group of order $2n$ and
the dicyclic group of order $4n$ are denoted as

$\displaystyle D_{2n}$ $\displaystyle=<a,b~{}|~{}a^{n}=b^{2}=e,bab=a^{-1}>,$
$\displaystyle Q_{n}$
$\displaystyle=<a,b~{}|~{}a^{2n}=e,b^{2}=a^{n},ab=ba^{-1}>.$

If $n$ is a power of $2,$ then $Q_{n}$ is called the _generalized quaternion
group_ of order $4n.$

###### Theorem 3.12

If $n$ is a prime power, then the $A_{\alpha}$-spectrum of
$\mathcal{P}(D_{2n})$ consists of the eigenvalue $\alpha n-1$ with
multiplicity $n-2$, the eigenvalue $\alpha$ with multiplicity $n-1$ and the
zeros of following polynomial

$(\alpha-x)\left(\alpha^{2}-2\alpha+2\alpha n^{2}+\alpha^{2}n-3\alpha
n-2\alpha nx-nx-n+x^{2}+2x+2\right)-(1-\alpha)^{2}n(\alpha+n-x-2).$

Proof. Consider the dihedral group
$D_{2n}=<a,b~{}|~{}a^{n}=b^{2}=e,bab=a^{-1}>$. Clearly, $<a>$ generates a
cyclic subgroup of order $n$, which is isomorphic to $\mathbb{Z}_{n}$. The
remaining $n$ elements of $D_{2n}$ form an independent set of
$\mathcal{P}(D_{2n})$, sharing the identity element $e$. Therefore, the
structure of the power graph of the dihedral group $D_{2n}$ can be obtained
from the power graph $\mathcal{P}(\mathbb{Z}_{n})$ by adding the $n$ pendent
vertices at the identity vertex $e.$ If $n=p^{z}$, where $z$ is positive
integer, then it is easy to see that

$\mathcal{P}(D_{2n})=P_{3}[K_{n-1},K_{1},\overline{K}_{n}],$

that is, the power graph $\mathcal{P}(D_{2n})$ of dihedral group for $n=p^{z}$
is a _pineapple_ graph (graph obtained from the complete graph by attaching
pendent vertices to any vertex of the complete graph). By using Theorem 2.3,
$\alpha n-1~{}\text{and}~{}\alpha$ are the $A_{\alpha}$-eigenvalues of
$\mathcal{P}(D_{2n})$ with multiplicities $n-2$ and $n-1$, respectively. The
remaining three $A_{\alpha}$-eigenvalues are given by the matrix

$\begin{pmatrix}n-2+\alpha&1-\alpha&0\\\
(1-\alpha)(n-2)&\alpha(2n-1)&(1-\alpha)n\\\ 0&1-\alpha&\alpha\\\
\end{pmatrix}.$

The following result gives the $A_{\alpha}$-spectrum of the generalized
quaternion group.

###### Theorem 3.13

If $n$ is a power of $2$, then the $A_{\alpha}$-spectrum of
$\mathcal{P}(Q_{n})$ is

$\\{4\alpha n-1,~{}(2\alpha
n-1)^{(2n-3)},~{}(4\alpha-1)^{(n)},(1+2\alpha)^{(n-2)},~{}x_{1},~{}x_{2},~{}x_{3}\\},$

where $x_{1},~{}x_{2},~{}x_{3}$ are the zeros of the characteristic polynomial
of the matrix given by (3.12).

Proof. Consider the quaternion group
$Q_{n}=<a,b~{}|~{}a^{2n}=e,b^{2}=a^{n},ab=ba^{-1}>$. By definition of power
graph, the identity element is always adjacent to every other vertex of
$\mathcal{P}(Q_{n})$. In particular, if $n$ is a power of $2$, then by one of
the observations in [20], we found that $a^{n}$ is also adjacent to all other
vertices of $\mathcal{P}(Q_{n})$. Thus, two vertices of the power graph
$\mathcal{P}(Q_{n})$ are connected to every other vertex of
$\mathcal{P}(Q_{n})$. Since $n$ copies of $K_{2}$ share the identity, so
$\mathcal{P}(Q_{n})$ can be written as

$\mathcal{P}(Q_{n})=K_{1,n+1}[K_{2},K_{2n-2},\underbrace{K_{2},K_{2},\dots,K_{2}}_{n}].$

By using Theorem 2.3, we see that
$\alpha_{1}=4n-2,\alpha_{2}=\alpha_{3}=\dots=\alpha_{n+2}=2$ and the
$A_{\alpha}$-spectrum of $\mathcal{P}(Q_{n})$ consists of the simple
eigenvalue $4\alpha n-1$, the eigenvalue $2\alpha n-1$ with multiplicity
$2n-3$, the eigenvalue $4\alpha-1$ with multiplicity $n$ and the remaining
$n+2$ eigenvalues are given by the following equitable quotient matrix

$M(Q_{n})=\begin{pmatrix}1+\alpha(4n-2)&(1-\alpha)(2n-2)&(1-\alpha)2&\dots&(1-\alpha)2\\\
(1-\alpha)2&2n-3+2\alpha&0&\dots&0\\\ (1-\alpha)2&0&1+2\alpha&\dots&0\\\
\vdots&\vdots&\vdots&\dots&\vdots\\\
(1-\alpha)2&0&0&\dots&1+2\alpha\end{pmatrix}.$ (3.11)

Now, applying Lemma 2.2 with
$X=\begin{pmatrix}1+\alpha(4n-2)&(1-\alpha)(2n-2)\\\
(1-\alpha)2&2n-3+2\alpha\end{pmatrix},~{}\beta=\begin{pmatrix}2(1-\alpha)\\\
0\end{pmatrix},B=(1+2\alpha)$ and $C=(0)$, we see that $1+2\alpha$ is the
$A_{\alpha}$ eigenvalue of the matrix (3.11) with multiplicity $n-1$. The
remaining three $A_{\alpha}$ eigenvalues of the matrix $M(Q_{n})$ in Equation
(3.11) are given by the following equitable quotient matrix

$\begin{pmatrix}1+\alpha(4n-2)&(1-\alpha)(2n-2)&2n(1-\alpha)\\\
2(1-\alpha)&2n-3+\alpha&0\\\ 2(1-\alpha)&0&1+2\alpha\end{pmatrix}.$ (3.12)

We raise the following questions about $A_{\alpha}$-spectrum of
$\mathcal{P}(D_{2n})$ and $\mathcal{P}(Q_{n})$.

###### Problem 3

Find the $A_{\alpha}$-spectrum of $\mathcal{P}(D_{2n})$ for the values of
$n\in\\{pq,pqr,p^{2}q,(pq)^{2}\\}$ and generalize (if possible) for any $n$.

###### Problem 4

Find the $A_{\alpha}$-spectrum of $\mathcal{P}(Q_{n})$ for the values of
$n\in\\{pq,pqr,p^{2}q,(pq)^{2}\\}$ and generalize (if possible) for any $n$.

Since for $\mathcal{G}=\mathbb{Z}_{n},~{}n=p,pq$, where $p<q$ are primes, the
multiplicity of $A_{\alpha}$ eigenvalue $\alpha n-1$ is exactly $\phi(n)$.
Thus the partial solution of Problem 1 for power graph
$\mathcal{P}(\mathbb{Z}_{n})$ are graphs $\mathcal{P}(\mathbb{Z}_{p})$ and
$\mathcal{P}(\mathbb{Z}_{pq})$. In order to show that there no other power
graphs for $\mathbb{Z}_{n}$, we have to show that $x-(\alpha n-1)$ is not the
factor of the characteristic polynomial of matrix (3.4). Similarly, in case of
generalized quaternion group
$Q_{n},~{}\text{for}~{}n=p^{z},~{}z\in\mathbb{N}$, multiplicity of $\alpha
n-1$ is $1=\phi(2)$ and equality does not hold for any other $n$. But as a
whole Problem 1 is still open.

## References

  * [1] J. Abawajy, A. Kelarev and M. Chowdhury, Power graphs: A survey, Electronic J. Graph Theory Appl. 1(2) (2013) 125–147.
  * [2] R. B. Bapat, S. J. Kirkland and S. Pati, The perturbed Laplacian matrix of a graph, Linear Multilinear Algebra, 49 (2001) 219–242.
  * [3] S. Banerjee and A. Adhikari, Signless Laplacian spectra of power graphs of certain finite groups, _AKCE Int. J. Graphs Comb._ DOI:10.1016/j.akcej.2019.03.009 (2019).
  * [4] A. E. Brouwer, W. H. Haemers Spectra of Graphs, Springer New York 2010.
  * [5] P. J. Cameron and S. Ghosh, The power graphs of a finite group, Dicrete Math. 311,13 (2011) 1220–1222.
  * [6] S. Chattopadhyay and P. Panigrahi, On Laplacian spectrum of power graphs of finite cyclic and dihedral groups, Linear Multilinear Algebra 63,7 (2015) 1345–1355.
  * [7] I. Chakrabarty, M. Ghosh and M. K. Sen, Undirected power graph of semigroups, Semigroup Forum 78 (2009) 410–426.
  * [8] T. T. Chelvan and M. Sattanathan, Power graphs of finite abelian groups, Algebra Discrete Math. 16,1 (2013) 33–41.
  * [9] D. M. Cvetković, M. Doob and H. Sachs, _Spectra of graphs. Theory and Applications,_ Pure and Applied Mathematics, 87. Academic Press, Inc. New York 1980.
  * [10] E. Fritscher and V. Trevisan, Exploring symmetries to decompose matrices and graphs preserving the spectrum, _SIAM J. Matrix Anal. Appl._ 37,1 (2016) 260–289.
  * [11] A. Hamzeh and A. R. Ashrafi, Spectrum and L-spectrum of the power graph and its main supergraph for certain finite groups _Filomat_ 31,16 (2017) 5323–5334.
  * [12] R. Horn and C. Johnson, Matrix Analysis Second Edition, Cambridge University Press 2013.
  * [13] A. V. Kelarev and S. J. Quinn, Directed graphs and combinatorial properties of semigroups, J.Algebra 251 (2002) 16–26.
  * [14] D Li, Y. Chen and J. Meng, The $A_{\alpha}$ spectral radius of trees and unicyclic graphs with given degree sequence, Appl. Math. Comp. 363 (2019) 124622
  * [15] Z. Mehranian, A. Gholami and A. R. Ashrafi, A note on the power graph of a finite group,_Int. J. of Group Theory_ 5 (2016) 1–10.
  * [16] Z. Mehranian, A. Gholami and A. R. Ashrafi, The spectra of power graphs of certain finite groups,_Linear Multilinear Algebra_ 65,5 (2016) 1003–1010.
  * [17] V. Nikiforov, Merging the A and Q spectral theories, Appl. Anal. Discrete Math. 11 (2017) 18–107.
  * [18] V. Nikiforov, G. Pasten, O. Rojo and R.L. Soto, On the $A_{\alpha}$ spectra of trees Linear Algebra Appl.520 (2017) 286–305.
  * [19] W. K. Nicholson, Introduction to abstract algebra, fourth edition, John Wiley and sons, New Jersey (2012).
  * [20] R. P. Panda, Laplacian spectra of power graphs of certain finite groups, Graphs and Comb. DOI:10.1007/s00373-019-02070-x (2019).
  * [21] S. Pirzada, _An Introduction to Graph Theory_ , Universities Press, Orient BlackSwan Hyderabad (2012).
  * [22] S. Pirzada, B. A. Rather, H. A. Ganie and R. Shaban, On $\alpha-$adjacency energy of graphs, communicated.
  * [23] B. A. Rather, S. Pirzada and Z. Goufei, On distance Laplacian spectra of power graphs of certain finite groups, preprint.
  * [24] C. Wang and S. Wang, The $A_{\alpha}$-spectra radii of graphs with given connectivity, _Mathematics_ 2019 7, 44; doi:10.3390/math7010044.