# On the solution set of semi-infinite tensor complementarity problem

R. Deba,1 and A. K. Dasb,2
_a Jadavpur University, Kolkata , 700 032, India._
_b Indian Statistical Institute, 203 B. T. Road, Kolkata, 700 108, India._
_1 Email<EMAIL_ADDRESS>
_2 Email<EMAIL_ADDRESS>

###### Abstract

In this paper, we introduce semi-infinite tensor complementarity problem to
provide an approach for considering a more realistic situation of the problem.
We prove the necessary and sufficient conditions for the existence of the
solution set. In this context, we study the error bounds of the solution set
in terms of residual function.

Keywords: Semi-infinite tensor complementarity problem, error bound, residual
function, $R_{0}$-tensor.

AMS subject classifications: 90C30, 90C33, 15A69.

11footnotetext: Corresponding author

##  Introduction

The tensor complementarity problem, TCP$(q,\mathcal{A})$, was introduced by
Song and Qi [37] and [36]. By reformulating the multilinear game as a tensor
complementarity problem, Huang and Qi [12] established a bridge between the
multilinear game and tensor complementarity problem. They demonstrated that
finding a Nash equilibrium point of the multilinear game is equivalent to
finding a solution to the resulting TCP.

Let $\mathcal{A}$ be a tensor of order $m$ and dimension $n,$ i.e.,
$\mathcal{A}\in T_{m,n}$ and a vector $q\in\mathbb{R}^{n},$ the tensor
complementarity problem denoted by TCP$(q,\mathcal{A})$ is to find
$x\in\mathbb{R}^{n}$ such that,

$q+\mathcal{A}x^{m-1}\geq 0,\;\;\;\;x\geq
0,\;\;\;\;\mbox{and}\;\;\;\;x^{T}(q+\mathcal{A}x^{m-1})=0.$ (1.1)

When the order of the tensor $m=2$ then the problem reduces to a linear
complementarity problem. Let $A$ be an $n\times n$ real matrix and a vector
$q\in\mathbb{R}^{n},$ the linear complementarity problem, denoted by
LCP$(q,A)$ is finding $x\in\mathbb{R}^{n}$ such that

$q+Ax\geq 0,\;\;\;\;x\geq 0,\;\;\;\;\mbox{and}\;\;\;\;x^{T}(q+Ax)=0.$ (1.2)

The idea of complementarity considers a large number of optimization problems.
The problems which can be constituted as linear complementarity problem
includes linear programming, linear fractional programming, convex quadratic
programming and the bimatrix game problem. It is well considered in the
literature on mathematical programming and occurs in a number of applications
in operations research, control theory, mathematical economics, geometry and
engineering. For recent works on this problem and applications see [18], [19],
[27], [25], [20], [6], [9], [14], [16], [33] and [23] references cited
therein.

The concept of PPT is originally motivated by the well-known linear
complementarity problem, and applied in many other settings. The PPT is
basically a transformation of the matrix of a linear system for exchanging
unknowns with the corresponding entries of the right hand side of the system.
For details see [5], [21], [32], [3] and [26].

Due to their prominence in scientific computing, complexity theory, and the
theoretical underpinnings of linear complementarity problems, a number of
matrix classes and their subclasses have received substantial study. For
recent work on this problem and applications see [13], [4], [10], [29], [31],
[7], [15]. For multivariate analysis and game problem, See [21], [17], [15],
[30], [28], [24] and references cited therein.

An implicit assumption shared by the nonlinear complementarity problem is that
the information about the mapping $F$ and the cone involved are all fixed and
completely independent of other related parameters. However, this type of
formulation is unable to model all realistic situations of the problem and
fails to explain the complete reality. For example, in optimal control or
engineering design fields [2], the data of the problem involves a time
parameter; in non-cooperative games (e.g., generalized Nash equilibrium [11]),
the strategy of each player is dependent on the strategy of the other players
in case of the realistic model. Here we introduce semi-infinite tensor
complementarity (SITCP), a generalized version of nonlinear omplementarity to
accommodate more number of realistic situations into the model.

The paper is organised as follows. Section 2 presents some basic notations and
results which are used in the next section. In section 3 we prove the
existence of the solution set for semi-infinite tensor complementarity problem
with some assumptions. We establish a connection between the solution set of
semi-infinite tensor complementarity problem and the solution sets of its
equivalent tensor complementarity problems. We establish the necessary and
sufficient conditions for the error bound of solution set to be level bounded
in terms of residual function.

##  Preliminaries

We begin by outlining the fundamental concepts and the notation that will be
applied throughout the text. Here we consider the vectors, matrices and
tensors of real entries. For any positive integer $n,$ the set $\\{1,...,n\\}$
is denoted by $[n]$ . Let $\mathbb{R}^{n}$ denote the $n$-dimensional
Euclidean space and $\mathbb{R}^{n}_{+}=\\{x\in\mathbb{R}^{n}:x\geq 0\\}.$ Any
vector $x\in\mathbb{R}^{n}$ is a column vector unless specified otherwise. The
Euclidean norm of a vector $x$ is defined as
$\|x\|_{2}=\sqrt{|x_{1}^{2}|+\cdots|x_{n}^{2}|}.$ The distance of $x$ from
$A\subseteq\mathbb{R}^{n}$ is denoted as dist$(x;A)$ and is defined as
dist$(x;A)=$ inf$\\{(x,a):a\in A\\}.$ The diameter of a set is denoted as
diam$A$ and is defined as diam$A=\sup_{x,y\in A}\|x-y\|.$ The unit ball in
$\mathbb{R}^{n}$ is $\mathbb{B}=\\{x\in\mathbb{R}^{n}:\|x\|\leq 1\\}.$ An
$m$th order $n$ dimensional real tensor $\mathcal{A}=(a_{i_{1}...i_{m}})$ is a
multidimensional array of entries $a_{i_{1}...i_{m}}\in\mathbb{R}$ where
$i_{j}\in[n]$ with $j\in[m]$. The set of all $m$th order $n$ dimensional real
tensors are denoted by $T_{m,n}.$ Shao [34] introduced a product of tensors.
Let $\mathcal{A}$ with order $q\geq 2$ and $\mathcal{B}$ with order $k\geq 1$
be two $n$-dimensional tensors. The product of $\mathcal{A}$ and $\mathcal{B}$
is a tensor $\mathcal{C}$ of order $(q-1)(k-1)+1$ and dimension $n$ with
entries
$c_{j\alpha_{1}\cdots\alpha_{m-1}}=\sum_{j_{2},\cdots,j_{m}\in[n]}a_{jj_{2}\cdots
j_{m}}b_{j_{2}\alpha_{1}}\cdots b_{j_{m}\alpha_{m-1}},$ where $j\in[n]$,
$\alpha_{1},\cdots,\alpha_{m-1}\in[n]^{k-1}.$

Then for a tensor $\mathcal{A}\in T_{m,n}$ and
$x\in\mathbb{R}^{n},\;\mathcal{A}x^{m-1}\in\mathbb{R}^{n}$ is a vector defined
by

$(\mathcal{A}x^{m-1})_{i}=\sum_{i_{2},...,i_{m}=1}^{n}a_{ii_{2}...i_{m}}x_{i_{2}}\cdots
x_{i_{m}},\;\forall\;i\in[n],$

and $\mathcal{A}x^{m}\in\mathbb{R}$ is a scalar defined by

$x^{T}\mathcal{A}x^{m-1}=\mathcal{A}x^{m}=\sum_{i_{1},...,i_{m}=1}^{n}a_{i_{1}...i_{m}}x_{i_{1}}\cdots
x_{i_{m}}.$

Given a vector $q\in\mathbb{R}^{n}$ and a tensor $\mathcal{A}\in T_{m,n}$ the
set of feasible solution of TCP$(q,\mathcal{A})$ is defined as
FEA$(q,\mathcal{A})=\\{x\in\mathbb{R}^{n}_{+}:q+\mathcal{A}x^{m-1}\geq 0\\}$
and the solution set of TCP$(q,\mathcal{A})$ as $S=$
SOL$(q,\mathcal{A})=\\{x\in$
FEA$(q,\mathcal{A}):x^{T}(q+\mathcal{A}x^{m-1})=0\\}.$ A residual function
$r(x)$ is a global (local) error bound for TCP if $\exists$ some constant
$c>0$ (and $\epsilon>0$) such that for each $x\in\mathbb{R}^{n}$ (when
$r(x)\leq\epsilon$)

$\mbox{dist}(x,S)\leq cr(x).$ (2.1)

We consider some definitions and results which are required for the next
sections.

######  definition 2.1:

[35], [8] The $i$th row subtensor of $\mathcal{A}=(a_{i_{1}...i_{m}})\in
T_{m,n}$ is denoted by $R_{i}(\mathcal{A})$ and its entries are given as
$(R_{i}(\mathcal{A}))_{i_{2}...i_{m}}=(a_{ii_{2}...i_{m}})$, where
$i_{j}\in[n]$ and $2\leq j\leq m.$

######  definition 2.2:

[39] A function $f:\mathbb{R}^{n}\mapsto\mathbb{R}^{n}$ is said to be level
bounded if for every $\alpha\geq 0$ the level set
$\\{x\in\mathbb{R}^{n}:f(x)\leq\alpha\\}$ is bounded.

######  definition 2.3:

[36] A tensor $\mathcal{A}\in T_{m,n}$ is said to be a $S$-tensor if the
system

$\mathcal{A}x^{m-1}>0,\;\;\;\;x>0$

has a solution.

######  definition 2.4:

[36] A tensor $\mathcal{A}\in T_{m,n}$ is said to be a $P$-tensor, if for each
$x\in\mathbb{R}^{n}\backslash\\{0\\}$, there exists an index $i\in[n]$ such
that ${}_{i}\neq 0$ and $x_{i}(\mathcal{A}x^{m-1})_{i}>0$.

######  definition 2.5:

[38], [36] A tensor $\mathcal{A}\in T_{m,n}$ is said to be a $R_{0}$-tensor if
the TCP$(0,\mathcal{A})$ has unique zero solution.

######  definition 2.6:

[41] For a given $\epsilon\geq 0$ a residual function $r(x)$ is said to be an
$\epsilon$-error bound for TCP if $\exists\,c>0$ such that dist$(x,S)\leq
cr(x)+\epsilon\;\;\;\;\forall\;x\in\mathbb{R}^{n}.$ If $\epsilon=0,$ the
definition reduces to the error bound.

######  theorem 2.1:

[22] A non-empty family $A$ of subsets of $\mathbb{R}^{n}$ is said to have the
finite intersection property (FIP) if the intersection over any finite
subcollection of $A$ is non-empty.

######  theorem 2.2:

[1] For a $P$-tensor $\mathcal{A}\in T_{m,n}$ and any $q\in\mathbb{R}^{n}$ the
solution set of TCP$(q,\mathcal{A})$ is nonempty and compact.

######  theorem 2.3:

[38] If $\mathcal{A}\in T_{m,n}$ is a an $R_{0}$-tensor then for
$q\in\mathbb{R}^{n},$ the solution set of the TCP$(q,\mathcal{A})$ is bounded.

##  Main results

We begin by introducing semi-infinite tensor complementarity problem
SITCP$(q(\omega),\mathcal{A}(\omega),\Omega)$.

Find a vector $x\in\mathbb{R}^{n}$ such that

$x\geq 0,\;\;F(x,\omega)\geq 0,\;\;x^{T}F(x,\omega)=0,\;\;\omega\in\Omega$
(3.1)

where $F:\mathbb{R}^{n}\times\Omega\mapsto\mathbb{R}^{n},$
$F=\mathcal{A}(\omega)x^{m-1}+q(\omega)$ and $\Omega$ is a set in
$\mathbb{R}^{p}.$ The solution set of
SITCP$(q(\omega),\mathcal{A}(\omega),\Omega)$ is denoted by $S^{*}=$
SOL$(q(\omega),\mathcal{A}(\omega),\Omega)=\\{x\geq
0:\mathcal{A}(\omega)x^{m-1}+q(\omega)\geq
0,\;x^{T}(\mathcal{A}(\omega)x^{m-1}+q(\omega))=0,\;\forall\;\omega\in\Omega\\}.$

Here we establish the necessary and sufficient conditions for $S^{*}$ to be
non-empty.

######  theorem 3.1:

Consider the SITCP$(q(\omega),\mathcal{A}(\omega),\Omega)$
and$\mathcal{A}(\omega_{0})$ is an $R_{0}$-tensor for some
$\omega_{0}\in\omega.$ $S^{*}\neq\phi$ if and only if
$\cap_{i=1}^{p}\mbox{SOL}(q(\omega_{i}),\mathcal{A}(\omega_{i}))\neq\phi$ for
finitely many points $\omega_{1},...,\omega_{p}\in\Omega.$

###### Proof.

If part: Since $S^{*}\subseteq\cap_{i=1}^{p}$
SOL$(q(\omega_{i}),\mathcal{A}(\omega_{i})),$ $S^{*}\neq\phi$ implies that
$\cap_{i=1}^{p}$ SOL$(q(\omega_{i})$, $\mathcal{A}(\omega_{i}))\neq\phi.$

Only if part: Since $\mathcal{A}(\omega_{0})$ is an $R_{0}$-tensor, by Theorem
2.3, the set SOL$(q(\omega_{0}),\mathcal{A}(\omega_{0}))$ is bounded. Which in
turn implies the boundedness of $S^{*}.$ On the other hand, since
SOL$(q(\omega),\mathcal{A}(\omega))$ is closed for each $\omega$, so is
$S^{*}.$ Thus $S^{*}$ is compact. By Theorem 2.1 of finite intersection of
compact sets, we obtain the result. ∎

In the next result we find the position of $S^{*}.$ For this purpose we
consider the followings:

(i) $(\mathcal{A}_{\max})_{i_{1}\cdots i_{m}}=(\bar{a}_{i_{1}\cdots
i_{m}})=\max_{\omega\in\Omega}a_{i_{1}\cdots i_{m}}(\omega)$
(ii) $(\mathcal{A}_{\min})_{i_{1}\cdots i_{m}}=(a^{\prime}_{i_{1}\cdots
i_{m}})=\min_{\omega\in\Omega}a_{i_{1}\cdots i_{m}}(\omega)$
(iii) $(q_{\max})i=\max_{\omega\in\Omega}q_{i}(\omega)$
(iv) $(q_{\min})i=\min_{\omega\in\Omega}q_{i}(\omega)$

######  theorem 3.2:

Consider SITCP$(q(\omega),\mathcal{A}(\omega),\Omega).$ If $\Omega$ is compact
and $\mathcal{A}(\omega)$ and $q(\omega)$ are continuous on $\Omega,$ then
$S^{*}\subseteq\mbox{SOL}(q_{\max},M_{\max})\cap\mbox{SOL}(q_{\min},M_{\min}).$
Furthermore, suppose in each row sub-tensor, $R_{i}(\mathcal{A}(\omega))$ and
in each row of $q(\omega),\;q_{i}(\omega)$ the minimum (and maximum) is
attained by a common $\omega^{\prime}$ (and $\bar{\omega}$), i.e., for each
$i=1,2,...,n,$ there exist $\omega^{\prime}_{i},\;\bar{\omega}_{i}\;\in\Omega$
such that $R_{i}(\mathcal{A}_{\min})=R_{i}(\mathcal{A}(\omega^{\prime}_{i}))$
, $(q_{\min})_{i}=q(\omega^{\prime}_{i})_{i}$ and
$R_{i}(\mathcal{A}_{\max})=R_{i}(\mathcal{A}(\bar{\omega}_{i})),$
$(q_{\max})_{i}=q(\bar{\omega}_{i})_{i}.$ Then

$S^{*}=\mbox{SOL}(q_{\max},\mathcal{A}_{\max})\cap\mbox{SOL}(q_{\min},\mathcal{A}_{\min}).$

###### Proof.

By the rules of maximization and minimization for the summation of functions
in Exercise 1.36 of [39], we have,

$\max_{\omega\in\Omega}(\mathcal{A}(\omega)x+q(\omega))\geq\mathcal{A}_{\max}x+q_{\max}$
(3.2)
$\min_{\omega\in\Omega}(\mathcal{A}(\omega)x+q(\omega))\leq\mathcal{A}_{\min}x+q_{\min}$
(3.3)

for all $x\geq 0.$ By using the inequalities (3.2) and (3.3) and following an
argument similar to that for Theorem 2.2 of [41], we obtain

$S^{*}\subseteq\mbox{SOL}(q_{\max},\mathcal{A}_{\max})\cap\mbox{SOL}(q_{\min},\mathcal{A}_{\min}).$
(3.4)

Now to prove the second part let us assume that the conditions of Theorem 3.2
hold. Then
$(\mathcal{A}_{\min}x^{m-1})_{i}=(\mathcal{A}(\omega^{\prime})x^{m-1})_{i}$
and
$(\mathcal{A}_{\min}x^{m-1})_{i}+(q_{\min})_{i}=(\mathcal{A}(\omega^{\prime}_{i})x^{m-1}+q(\omega_{i}))_{i}.$
Therefore SOL$(q_{\min},\mathcal{A}_{\min})$
=SOL$(q(\omega^{\prime}_{i}),\mathcal{A}(\omega^{\prime}_{i})).$ Similarly we
have SOL $(q_{\max},$ $\mathcal{A}_{\max})$ =SOL
$(q(\bar{\omega}_{i}),\mathcal{A}(\bar{\omega}_{i})).$ Since
$S^{*}=\cap_{\omega\in\Omega}\mbox{SOL}(q(\omega),\mathcal{A}(\omega)),$ using
(3.2) and (3.3) we conclude that

$S^{*}=\mbox{SOL}(q_{\max},\mathcal{A}_{\max})\cap\mbox{SOL}(q_{\min},\mathcal{A}_{\min}).$

∎

Here we provide an example to illustrate the result of Theorem 3.2.

######  example 3.1:

Let $\mathcal{A}(\omega)\in T_{3,2}$ and $q(\omega)\in\mathbb{R}^{2}$ be such
that
$a_{111}=(1-2\omega^{3}),\;a_{121}=1-\omega,\;a_{112}=(1-\omega),\;a_{122}=-1,\;a_{211}=a_{212}=a_{221}=0\;a_{222}=-\omega^{2}\;$
and $q(\omega)=\left(\begin{array}[]{c}1\\\ \omega^{2}\end{array}\right)$ and
$\omega\in\Omega=[0,1].$ Then
$\mathcal{A}(\omega)x^{2}=\left(\begin{array}[]{c}(1-2\omega^{3})x_{1}^{2}+2(1-\omega)x_{1}x_{2}-x_{2}^{2}\\\
-\omega^{2}x_{2}^{2}\end{array}\right).$ Now consider the
SITCP$(q(\omega),\mathcal{A}(\omega),\Omega)$ which is to find
$x=\left(\begin{array}[]{c}x_{1}\\\ x_{2}\end{array}\right)\in\mathbb{R}^{2}$
such that

$x_{1}\geq 0;\;\;\;\;(\mathcal{A}(\omega)x^{2})_{1}\geq
0;\;\;\;\;x_{1}[(1-2\omega^{3})x_{1}^{2}+2(1-\omega)x_{1}x_{2}-x_{2}^{2}+1]=0,$
(3.5) $x_{2}\geq 0;\;\;\;\;(\mathcal{A}(\omega)x^{2})_{2}\geq
0;\;\;\;\;x_{2}[-\omega^{2}x_{2}^{2}+\omega^{2}]=0.$ (3.6)

Solving the equations (3.5) and (3.6) we obtain the solution set for
$\omega\in\Omega,$ which is $\left\\{\left(\begin{array}[]{c}0\\\
0\end{array}\right),\;\left(\begin{array}[]{c}0\\\
1\end{array}\right),\;\left(\begin{array}[]{c}\frac{2(1-\omega)}{1-2\omega^{3}}\\\
1\end{array}\right)\right\\}.$ For $\omega=1$ we get
$\frac{2(1-\omega)}{1-2\omega^{3}}=0$ so
$S^{*}=\left\\{\left(\begin{array}[]{c}0\\\
0\end{array}\right),\;\left(\begin{array}[]{c}0\\\
1\end{array}\right)\right\\}.$

Now for the given tensor $\mathcal{A}(\omega),$ let
$\mathcal{A}_{\max}=(\bar{a}_{ijk})\in T_{3,2}.$ Then we obtain
$\bar{a}_{111}=1,\;\bar{a}_{121}=\bar{a}_{112}=1,\;\bar{a}_{122}=-1$ and
$\bar{a}_{211}=\bar{a}_{212}=\bar{a}_{221}=\bar{a}_{222}=0\;$ and
$q_{max}=\left(\begin{array}[]{c}1\\\ 1\end{array}\right).$ Then the
TCP$(q_{\max},\mathcal{A}_{\max})$ which is finding
$x=\left(\begin{array}[]{c}x_{1}\\\ x_{2}\end{array}\right)\in\mathbb{R}^{2}$
such that

$x_{1}\geq 0;\;\;\;\;(\mathcal{A}_{\max}x^{2})_{1}\geq
0;\;\;\;\;x_{1}[x_{1}^{2}+2x_{1}x_{2}-x_{2}^{2}+1]=0,$ (3.7) $x_{2}\geq
0;\;\;\;\;(\mathcal{A}_{max}x^{2})_{2}\geq 0;\;\;\;\;x_{2}[0+1]=0.$ (3.8)

Solving (3.7) and (3.8) we get
SOL$(q_{\max},\mathcal{A}_{\max})=\left\\{\left(\begin{array}[]{c}0\\\
0\end{array}\right)\right\\}.$

Again, let $\mathcal{A}_{\min}=(a^{\prime}_{ijk})\in T_{3,2}.$ Then we obtain
$a^{\prime}_{111}=-1,\;a^{\prime}_{121}=a^{\prime}_{112}=0,\;a^{\prime}_{122}=-1$
and
$a^{\prime}_{211}=a^{\prime}_{212}=a^{\prime}_{221}=0,\;a^{\prime}_{222}=-1\;$
and $q_{min}=\left(\begin{array}[]{c}1\\\ 0\end{array}\right).$ Then
TCP$(q_{\min},\mathcal{A}_{\min})$ is finding
$x=\left(\begin{array}[]{c}x_{1}\\\ x_{2}\end{array}\right)\in\mathbb{R}^{2}$
such that

$x_{1}\geq 0;\;\;\;\;(\mathcal{A}_{\min}x^{2})_{1}\geq
0;\;\;\;\;x_{1}[-x_{1}^{2}+-x_{2}^{2}+1]=0,$ (3.9) $x_{2}\geq
0;\;\;\;\;(\mathcal{A}_{min}x^{2})_{2}\geq 0;\;\;\;\;x_{2}[-x_{2}^{2}]=0.$
(3.10)

Solving (3.9) and (3.10) we have
SOL$(q_{\min},\mathcal{A}_{\min})=\left\\{\left(\begin{array}[]{c}0\\\
0\end{array}\right),\;\left(\begin{array}[]{c}1\\\
0\end{array}\right)\right\\}.$

Thus $S^{*}\supset$ SOL$(q_{\max},\mathcal{A}_{\max})\cap$
SOL$(q_{\min},\mathcal{A}_{\min}),$i.e., the inclusion is strict.

Now we replace $q(\omega)$ by $\bar{q}=\left(\begin{array}[]{c}1\\\
1\end{array}\right).$ Then the SITCP$(\bar{q},\mathcal{A}(\omega),\Omega)$
which is to find $x=\left(\begin{array}[]{c}x_{1}\\\
x_{2}\end{array}\right)\in\mathbb{R}^{2}$ such that

$x_{1}\geq 0;\;\;\;\;(\mathcal{A}(\omega)x^{2})_{1}\geq
0;\;\;\;\;x_{1}[(1-2\omega^{3})x_{1}^{2}+2(1-\omega)x_{1}x_{2}-x_{2}^{2}+1]=0,$
(3.11) $x_{2}\geq 0;\;\;\;\;(\mathcal{A}(\omega)x^{2})_{2}\geq
0;\;\;\;\;x_{2}[-\omega^{2}x_{2}^{2}+1]=0.$ (3.12)

Solving the equations (3.11) and (3.12) we obtain the solution set for
$\omega\in\Omega,$ which is $\left\\{\left(\begin{array}[]{c}0\\\
0\end{array}\right),\;\left(\begin{array}[]{c}\alpha(\omega)\\\
\frac{1}{\omega}\end{array}\right)\right\\}.$ Here $\alpha(\omega)$ is the
positive root of the equation

$\omega^{2}(1-2\omega^{3})x_{1}^{2}-2\omega(1-\omega)x_{1}-(1-\omega^{2})=0.$

For $\omega=1$ we get $\alpha(1)=0.$ Therefore
$S^{*}=\left\\{\left(\begin{array}[]{c}0\\\ 0\end{array}\right)\right\\}.$

Then TCP$(q_{\max},\mathcal{A}_{\max})=$ TCP$(\bar{q},\mathcal{A}_{\max}).$
SOL$(\bar{q},\mathcal{A}_{\max})=\left\\{\left(\begin{array}[]{c}0\\\
0\end{array}\right)\right\\}.$

Now, TCP$(\bar{q},\mathcal{A}_{\min})$ is finding
$x=\left(\begin{array}[]{c}x_{1}\\\ x_{2}\end{array}\right)\in\mathbb{R}^{2}$
such that

$x_{1}\geq 0;\;\;\;\;(\mathcal{A}_{\min}x^{2})_{1}\geq
0;\;\;\;\;x_{1}[-x_{1}^{2}+-x_{2}^{2}+1]=0,$ (3.13) $x_{2}\geq
0;\;\;\;\;(\mathcal{A}_{min}x^{2})_{2}\geq 0;\;\;\;\;x_{2}[-x_{2}^{2}+1]=0.$
(3.14)

Solving (3.13) and (3.14) we have
SOL$(\bar{q},\mathcal{A}_{\min})=\left\\{\left(\begin{array}[]{c}0\\\
0\end{array}\right),\;\left(\begin{array}[]{c}1\\\
0\end{array}\right),\;\left(\begin{array}[]{c}0\\\
1\end{array}\right)\right\\}.$ In this case we have
$S^{*}=SOL(q_{\max},\mathcal{A}_{\max})\cap SOL(q_{\min},\mathcal{A}_{\min}).$

For the next result we define semi-infinite $S$-tensor.

######  definition 3.7:

A tensor $\mathcal{A}(\omega)\in T_{m,n}$ is said to be a semi-infinite
$S$-tensor with respect to the set $\omega$ if $\exists$ a vector $x>0$ such
that $\mathcal{A}(\omega)x^{m-1}>0,\;\forall\;\omega\in\Omega.$

Here we establish a connection between the semi-infinite $S$-tensor and the
feasibility of SITCP$(q(\omega),\mathcal{A}(\omega),\Omega).$

######  theorem 3.3:

Consider the SITCP$(q(\omega),\mathcal{A}(\omega),\Omega)$ where $\Omega$ is
compact and all the elements of $\mathcal{A}(\omega)$ are continuous on
$\Omega.$ Then $\mathcal{A}(\omega)$ is a semi-infinite $S$-tensor relative to
$\Omega$ if and only if SITCP$(q(\omega),\mathcal{A}(\omega),\Omega)$ is
feasible for all $q(\omega)\in C(\Omega)$ where $C(\Omega)$ denotes all
continuous mapping on $\Omega.$

###### Proof.

If part: Since $\mathcal{A}(\omega)$ is a semi-infinite $S$-tensor, $\exists$
$x>0$ such that $\mathcal{A}(\omega)x^{m-1}>0.$ Then $\exists$ a sufficiently
small scalar $\lambda>0$ such that $\mathcal{A}(\omega)x^{m-1}\geq\lambda
e>0,$ for all $\omega\in\Omega$ where $e=(1,\cdots,1)^{T}.$ Now choose
$\alpha>0$ with $\alpha e>-q_{\min}.$ Choosing
$\bar{\alpha}=\left(\frac{\alpha}{\lambda}\right)^{\frac{1}{m-1}}>0$ we have
$\bar{\alpha}x>0.$ Also,
$\mathcal{A}(\omega)(\bar{\alpha}x)^{m-1}$=$\frac{\alpha}{\lambda}\mathcal{A}(\omega)x^{m-1}\geq\alpha
e>-q_{\min}.$ Thus for $\bar{\alpha}x>0$ we have
$\mathcal{A}(\omega)(\bar{\alpha}x)^{m-1}+q(\omega)\geq\mathcal{A}(\omega)(\bar{\alpha}x)^{m-1}+q_{\min}>0.$
Therefore, $\bar{\alpha}x$ is a feasible point of
SITCP$(q(\omega),\mathcal{A}(\omega),\Omega)$.

Only if part: Let $q(\omega):=\tilde{q}<0$ for all $\omega\in\Omega.$ Let the
SITCP$(\tilde{q},$ $\mathcal{A}(\omega),\Omega)$ be feasible. Then $\exists$ a
vector $x\geq 0$ such that $\mathcal{A}(\omega)x^{m-1}+\tilde{q}\geq
0\implies\mathcal{A}(\omega)x^{m-1}\geq-\tilde{q}>0$ for all
$\omega\in\Omega$. Since $\mathcal{A}(\omega)x^{m-1}$ is continuous on
$\Omega$ there exists a sufficiently small $\lambda>0$ such that $x+\lambda
e>0,$ and $\mathcal{A}(\omega)(x+\lambda e)^{m-1}>0.$ ∎

The following corollary provides a necessary condition for feasibility of the
solution set of SITCP$(q(\omega),\mathcal{A}(\omega),\Omega).$

######  corollary 3.1:

Consider the SITCP$(q(\omega),\mathcal{A}(\omega),\Omega).$ Suppose $\Omega$
is compact and $\mathcal{A}(\omega)$ is continuous on $\Omega.$ If
$\mathcal{A}_{\min}$ is an $S$-tensor, then
SITCP$(q(\omega),\mathcal{A}(\omega),\Omega)$ is feasible for all
$q(\omega)\in C(\Omega).$

###### Proof.

Let $\mathcal{A}_{\min}$ be an $S$-tensor. Then for some $x>0$ we have
$\mathcal{A}_{\min}x^{m-1}>0.$ From the definition of $\mathcal{A}_{\min}$ it
follows that $\mathcal{A}(\omega)x^{m-1}>0$ for all $\omega\in\Omega.$ This
implies that $\mathcal{A}(\omega)$ is a semi-infinite $S$-tensor. Hence by
Theorem 3.3, we conclude that SITCP$(q(\omega),\mathcal{A}(\omega),\Omega)$ is
feasible for all $q(\omega)\in C(\Omega).$ ∎

Now we establish an $\epsilon-$error bound for the solution set of semi-infnte
tensor complementarity problem. Here $\epsilon$ represents the degree of
approximation of the set $S^{*}$

######  theorem 3.4:

Consider the SITCP$(q(\omega),\mathcal{A}(\omega),\Omega).$ Suppose the
solution set $S^{*}$ is nonempty. If $\mathcal{A}(\omega_{0})$ is an
$P$-tensor for some $\omega_{0}\in\Omega,$ then there exist $c>0$ and
$\epsilon>0$ with
$\epsilon\leq\mbox{diam}(\mbox{SOL}(q(\omega_{0}),\mathcal{A}(\omega_{0}))$
such that

$\mbox{dist}(x,S^{*})\leq c\;r(x)+\epsilon$

where residual function is
$r_{y}(x)=\max_{\omega\Omega}\|\min\\{x,[\mathcal{A}(\omega)(x-y)]^{\frac{1}{m-1}}+[\mathcal{A}(\omega)y^{m-1}]^{\frac{1}{m-1}}\\}\|,$
and $y\in(SOL(q(\omega_{0}),\mathcal{A}(\omega_{0})).$

###### Proof.

Since $\mathcal{A}(\omega_{0})$ is an $P$-tensor,
SOL$(q(\omega_{0}),\mathcal{A}(\omega_{0}))$ is bounded. This implies
$\exists$ an $\epsilon>0$ such that
SOL$(q(\omega_{0}),\mathcal{A}(\omega_{0}))\subseteq
S^{*}+\epsilon\mathbb{B}.$ Consequently,

$\mbox{dist}(x,S^{*})\leq\mbox{dist}(x,SOL(q(\omega_{0}),\mathcal{A}(\omega_{0})))+\epsilon,\forall
x\in\mathbb{R}^{n}.$ (3.15)

Notice that $S^{*}\subseteq$ SOL$(q(\omega_{0}),\mathcal{A}(\omega_{0})).$
Therefore the diameter of the set SOL$(q(\omega_{0}),\mathcal{A}(\omega_{0}))$
is an upper bound of $\epsilon.$ By Theorem 3.2 of [40] for the tensor
complementarity problem TCP$(q(\omega_{0}),\mathcal{A}(\omega_{0})),$ there
exists $c>0$ such that $\forall\;x\in\mathbb{R}^{n},$

$\mbox{dist}(x,SOL(q(\omega_{0}),\mathcal{A}(\omega_{0})))\leq
c\|\min\\{x,[\mathcal{A}(\omega_{0})(x-y)]^{\frac{1}{m-1}}+[\mathcal{A}(\omega_{0})y^{m-1}]^{\frac{1}{m-1}}\\}\|.$
(3.16)

From (3.15) and (3.16) the desired result follows. ∎

For the next result we define semi-infinite $R_{0}$-tensor and establish a
connection between $R_{0}$-tensor and semi-infinite $R_{0}$-tensor.

######  definition 3.8:

The tensor $\mathcal{A}(\omega)$ is said to be a semi-infinite $R_{0}$-tensor
relative to a set $\Omega$ if the SITCP$(0,\mathcal{A}(\omega),\Omega)$ has
zero as its unique solution, i.e.,

$x\geq 0,\;\;\mathcal{A}(\omega)x\geq
0,\;\;x^{T}\mathcal{A}(\omega)x=0,\;\;\;\;\forall\omega\in\Omega\implies x=0.$
(3.17)

######  theorem 3.5:

Consider the SITCP$(q(\omega),\mathcal{A}(\omega),\Omega).$ If
$\mathcal{A}(\omega_{0})$ is an $R_{0}$-tensor for some $\omega_{0}\in\Omega.$
Then $\mathcal{A}(\omega)$ is a semi-infinite $R_{0}$-tensor relative to
$\Omega.$

###### Proof.

Since $\mathcal{A}(\omega_{0})$ is an $R_{0}$-tensor, we have
SOL$(0,\mathcal{A}(\omega_{0}))=\\{0\\}.$ Also,
$S^{*}\subseteq\cap_{\omega\in\Omega}$ SOL$(0,\mathcal{A}(\omega),\Omega)$
Therefore $S^{*}=\\{0\\}.$ Hence the result. ∎

Here we prove the necessary and sufficient conditions for the error bound of
solution set to be level bounded in terms of residual function. To solve the
semi-infinite tensor complementarity problem is equivalent to finding
$x\in\mathbb{R}^{n}$ such that $x\in\mbox{SOL}(q(\omega),\mathcal{A}(\omega))$
for all $\omega\in\Omega.$ However, in many cases, it is possible to obtain
$x\in\mbox{SOL}(q(\omega),\mathcal{A}(\omega))$ for some $\omega.$ In this
case, it is important to provide a quantitative measure of the closeness of
each $x\in\mathbb{R}^{n}$ to each individual set
SOL$(q(\omega),\mathcal{A}(\omega))$ in terms of some residual functions
$r(x)$. In other words, we find $c>0$ such that

$\mbox{dist}(x,\mbox{SOL}(q(\omega),\mathcal{A}(\omega)))\leq
cr(x),\;\forall\;\omega\in\Omega,\;\forall\;x\in\mathbb{R}^{n}$

which is equivalent to

$\max_{\omega\in\Omega}\mbox{dist}(x,\mbox{SOL}(q(\omega),\mathcal{A}(\omega)))\leq
cr(x),\;\;\forall\;x\in\mathbb{R}^{n}.$

Here $c$ is said to be weak error bound. The importance of weak error bound is
that the solution $S^{*}$ is need not be nonempty as required in case of error
bound.

######  theorem 3.6:

Consider the SITCP$(q(\omega),\mathcal{A}(\omega),\Omega).$ Suppose $\Omega$
is compact and $\mathcal{A}(\omega)$ and $q(\omega)$ are continuous. Then the
residual function,
$r(x)=\max_{\omega\in\Omega}\|\min(x,\mathcal{A}(\omega)x+q(\omega))\|^{2}$ is
level bounded if and only if the tensor $\mathcal{A}(\omega)$ is a semi-
infinite $R_{0}$-tensor relative to $\Omega.$

###### Proof.

Only if part: We prove the first result by contrapositive method. By this
approach we first assume that $r(x)$ is not level bounded. Then $\exists$ a
sequence $\\{x_{n}\\}\mapsto\infty$ as $n\mapsto\infty,$ $\\{r(x_{n})\\}$ is
bounded. We assume that $\frac{x_{n}}{\|x_{n}\|^{m-1}}$ converge to the limit
$x_{0}$ with $\|x_{0}\|=1.$ Taking into account the continuity of $q(\omega)$
and $\mathcal{A}(\omega)$ and the compactness of $\Omega$, we see that $r(x)$
is continuous and $q(\omega)$ is bounded on $\Omega.$ Hence,
$\lim_{n\mapsto\infty}\frac{r(x_{n})}{\|x_{n}\|^{m-1}}=0$ and
$\lim_{n\mapsto\infty}\frac{q(\omega)}{\|x_{n}\|^{m-1}}=0$ for all
$\omega\in\Omega.$ Now

$\frac{r(x_{n})}{\|x_{n}\|^{2(m-1)}}=\max_{\omega\in\Omega}\|\min\left\\{\frac{x_{n}}{\|x_{n}\|^{m-1}},\frac{\mathcal{A}(\omega)x+q(\omega)}{\|x_{n}\|^{m-1}}\right\\}\|^{2}.$
(3.18)

Taking limit of both sides of the equation (3.18) as $n\mapsto\infty$ we
obtain,

$\max_{\omega\in\Omega}\|\min\left\\{x_{0},\mathcal{A}(\omega)x_{0}^{m-1}\right\\}\|^{2}=0.$

This means that a nonzero vector $x_{0}$ is a solution of
SITCP$(0,\mathcal{A}(\omega),\Omega).$ Hence $\mathcal{A}(\omega)$ is not a
semi-infinite $R_{0}$-tensor. This completes the proof.

If part: Suppose on the contrary that the
SITCP$(0,\mathcal{A}(\omega),\Omega)$ has a nonzero vector $x$ as a solution.
Let $I(x)=\\{i:x_{i}=0\\}$ and $J(x)=\\{i:x_{i}>0\\}.$ The compactness of
$\Omega$ and the continuity of $q$ ensures that $q(\omega)$ is bounded on
$\Omega.$ Thus there exists a scalar $K>0$ such that, for any $k\geq K,$

$kx_{i}\geq q_{i}(\omega)\mbox{ for all }\omega\in\Omega\mbox{ and }i\in
J(x).$ (3.19)

Given any $k\geq K,$ we have

$\displaystyle r(kx)$
$\displaystyle=\max_{\omega\in\Omega}\|\min(kx,k^{m-1}\mathcal{A}(\omega)x^{m-1}+q(\omega))\|^{2}$
(3.20)
$\displaystyle\leq\sum_{i=1}^{n}\max_{\omega\in\Omega}\\{\min(kx_{i},(k^{m-1}\mathcal{A}(\omega)x^{m-1})_{i}+q_{i}(\omega))\\}^{2}$

Now we consider the following cases.

Case-1. We consider the case when $i\in J(x).$ Then
$((\mathcal{A}(\omega)x^{m-1})_{i}=0.$ It follows from (3.19) that

$\max_{\omega\in\Omega}\\{\min\\{kx_{i},k^{m-1}(\mathcal{A}(\omega)x^{m-1})_{i}+q_{i}(\omega))\\}\\}^{2}=\max_{\omega\in\Omega}q_{i}(\omega)^{2}.$
(3.21)

Case-2. We consider the case when $i\in I(x).$ If
$k^{m-1}(\mathcal{A}(\omega)x^{m-1})_{i}+q_{i}(\omega)\geq 0,$ we have

$\\{\min(kx_{i},k^{m-1}(\mathcal{A}(\omega)x^{m-1})_{i}+q_{i}(\omega))\\}^{2}=0.$
(3.22)

If $k^{m-1}(\mathcal{A}(\omega)x^{m-1})_{i}+q_{i}(\omega)<0,$ then by the fact
$q_{i}(\omega)\leq k^{m-1}(\mathcal{A}(\omega)x^{m-1})_{i}+q_{i}(\omega)<0$ we
obtain

$\\{\min(kx_{i},k^{m-1}(\mathcal{A}(\omega)x^{m-1})_{i}+q_{i}(\omega))\\}^{2}\leq
q_{i}(\omega)^{2}.$ (3.23)

Thus combining (3.22) and (3.23) we have

$\max_{\omega\in\Omega}[\min(kx_{i},k^{m-1}(\mathcal{A}(\omega)x^{m-1})_{i}+q_{i}(\omega))]^{2}\leq\max_{\omega\in\Omega}q_{i}(\omega)^{2}.$
(3.24)

Putting the facts (3.20), (3.21) and (3.24) together, it follows that

$r(kx)\leq\sum_{i=1}^{n}\max_{\omega\in\Omega}q_{i}(\omega)^{2}<\infty$

for all $k\geq K.$ This contradicts the level boundedness of $r(x).$ ∎

##  Conclusion

In this paper we introduce semi-infinite tensor complementarity problem to
accommodate more realistic situation of the problem. We show that the solution
set of semi-infinite tensor complementarity problem exists with some
assumption. An example is illustrated in detail to establish the result. We
establish a connection between semi-infinite tensor and its equivalent tensors
to obtain the solution of semi-infinite tensor complementarity problem.
Finally, we show that the error bound of the solution set is level bounded in
terms of residual function.

##  Acknowledgment

The author R. Deb is grateful to the Council of Scientific $\&$ Industrial
Research (CSIR), India, for providing financial assistance through the Junior
Research Fellowship program.

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# Finite groups and the class-size prime graph revisited

V. Sotomayor  Instituto Universitario de Matemática Pura y Aplicada (IUMPA-
UPV), Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia,
Spain.
🖂<EMAIL_ADDRESS>

This research is supported by Proyecto PGC2018-096872-B-I00 (MCIU / AEI /
FEDER, UE), and by Proyectos AICO/2020/298 and CIAICO/2021/163, Generalitat
Valenciana (Spain). Additionally, the author would like to thank S. Dolfi, E.
Pacifici and L. Sanus for introducing him to the research on the class-size
prime graph.

###### Abstract

We report on recent progress concerning the relationship that exists between
the algebraic structure of a finite group and certain features of its class-
size prime graph.

##  1 Introduction

Hereafter, all groups considered are finite. In recent decades, the influence
on the structure of a finite group $G$ of the arithmetical properties of its
set $cs(G)$ of conjugacy class sizes has been a widely investigated area. In
order to better understand the arithmetical structure of that set of positive
integers, the use of the prime graph built on the class sizes has been gaining
an increasing interest. In general, the prime graph $\Delta(X)$ built on a set
of positive integers $X$ is the (simple undirected) graph whose vertex set
consists of the prime divisors of the numbers in $X$, and the edge set
contains pairs $\\{p,q\\}$ of vertices such that the product $pq$ divides some
number in $X$. In particular, when $X=cs(G)$, we briefly write $\Delta(G)$ for
the class-size prime graph, $V(G)$ for the vertex set, and $E(G)$ for the edge
set.

The relationship that exists between graph-theoretical properties of
$\Delta(G)$ and the structure of $G$ itself has attracted the interest of many
authors. In this setting, two main questions naturally arise: what can be said
about the structure of $G$ if some information on $\Delta(G)$ is known, and
which graphs can occur as $\Delta(G)$ for some group $G$? We usually refer to
the expository paper [14] due to Camina and Camina for an overview of results
on this framework. Besides, the survey [35] of Lewis, which is mainly devoted
to summarise results regarding graphs associated with character degrees, has
also a section focused on graphs built on conjugacy class sizes. Nevertheless,
the aforementioned papers were publised in 2011 and 2008, respectively, and
numerous interesting results have appeared since then. Thus, our aim in this
note is to bring them together in one place, providing references to the
literature for their proofs. For the sake of completeness, the corresponding
results that were already gathered in [14] or [35] will be again presented
here.

The structure of the paper is mostly based on the two above mentioned
questions. More concretely, in Section 2 we focus on how graph-theoretical
properties of $\Delta(G)$ are reflected and influenced by the algebraic
structure of $G$. On the other hand, in Section 3 we pay attention to the
graphs that may appear as $\Delta(G)$ for some group $G$. In the last section,
we review variations of these results, which consider the class-size prime
graph built on some smaller subsets of elements of the group, as $p$_-regular_
elements, $p$_-singular_ elements, _real_ elements, _vanishing_ elements,
elements lying in a normal subgroup, and elements that belong to the factors
of a factorised group.

Notation: for a finite group $G$, let $x^{G}=\\{g^{-1}xg\;:\;g\in G\\}$ be the
conjugacy class of an element $x\in G$, and let $|x^{G}|$ be its size. Recall
that, in virtue of the orbit-stabiliser theorem,
$|x^{G}|=|G:\boldsymbol{\operatorname{C}}_{G}(x)|$. As previously said,
$cs(G)=\\{|x^{G}|\;:\;x\in G\\}$. By $\operatorname{Irr}(G)$ we mean the set
of all irreducible complex characters of the group $G$. The sets of all Sylow
$p$-subgroups and Hall $\pi$-subgroups of $G$ are
$\operatorname{Syl}_{p}\left(G\right)$ and
$\operatorname{Hall}_{\pi}\left(G\right)$, respectively, where $p$ is a prime
number and $\pi$ is a set of primes. The set of all prime divisors of the
order of $G$ is $\pi(G)$. The semidirect product of $A$ and $B$, where $A$ is
normalised by $B$, is written by $A\rtimes B$. Finally, we use CFSG to denote
the classification of finite simple groups. The remaining notation and
terminology is standard within Finite Group Theory.

##  2 The interplay between $\Delta(G)$ and the structure of $G$

Perhaps the earliest result in this strand of research is the following one
due to Sylow in 1872: if $cs(G)$ only contains powers of a fixed prime $p$,
then the centre of $G$ is non-trivial. This fact, that elementarily follows
from the class equation, can actually be rephrased in terms of the class-size
prime graph:

######  Lemma 2.1.

_([41], rephrased)._ Let $G$ be a group such that $\Delta(G)$ is an isolated
vertex. Then $\boldsymbol{\operatorname{Z}}(G)$ is non-trivial.

In fact, Baer in 1953 proved that all the class sizes of prime power order
elements of a group are powers of a fixed prime $p$ if and only if $G$ has a
central $p$-complement (cf. [3, Proposition 1]). Thus, we have the next
corollary.

######  Corollary 2.2.

Let $G$ be a group. Then $\Delta(G)$ is an isolated vertex $p$ if and only if
$G=P\times H$ with $P\in\operatorname{Syl}_{p}\left(G\right)$ and
$H\leqslant\boldsymbol{\operatorname{Z}}(G)$.

For the “dual” condition on the class sizes, i.e. a fixed prime $p$ does not
divide any class size of a group $G$, it is not difficult to prove that this
happens if and only if $G$ has a central Sylow $p$-subgroup. In other words,
we have a characterisation of the absence of a vertex in the class-size prime
graph:

######  Lemma 2.3.

Let $G$ be a group, $p\in\pi(G)$ and
$P\in\operatorname{Syl}_{p}\left(G\right)$. Then $p$ does not lie in $V(G)$ if
and only if $P\leqslant\boldsymbol{\operatorname{Z}}(G)$.

In this spirit, the absence of an edge in $\Delta(G)$ has also a strong impact
on the group structure. This fact was studied in 1953 by Itô, but without any
mention of a graph. More concretely, he proved in [34, Proposition 5.1] that
whenever $p\neq q$ are primes that divide two distinct conjugacy class sizes
of a group $G$, and $pq$ does not divide any class size of $G$, then such a
group has either a normal $p$-complement or a normal $q$-complement. It also
follows from Itô’s proof that the Sylow $p$-subgroups (or $q$-subgroups) are
abelian, although he did not explicitly stated it. Indeed, Dolfi obtained in
[20, Theorem 13] that both the Sylow $p$-subgroups and Sylow $q$-subgroups are
also abelian if we assume the solubility of the group. However, Casolo and
Dolfi moved beyond that, obtaining statement (ii) of the following theorem.

######  Theorem 2.4.

Let $G$ be a finite group. Suppose that $p\neq q$ are non-adjacent vertices of
$\Delta(G)$. Then the following conclusions hold:

  * _(a)_

_([34, Proposition 5.1])._ $G$ has, up to interchanging $p$ and $q$, a normal
$p$-complement and abelian Sylow $p$-subgroups.

  * _(b)_

_([16, Theorem B])._ $G$ is $\\{p,q\\}$-soluble of $\\{p,q\\}$-length at most
1, and both the Sylow $p$-subgroups and Sylow $q$-subgroups of $G$ are
abelian.

It is worth mentioning that the proof of the $\\{p,q\\}$-solubility of $G$
uses the CFSG via the next theorem.

######  Theorem 2.5.

_([16, Theorem 9])._ Let $G$ be a finite group with
$\boldsymbol{\operatorname{F}}(G)=1$. Then $\Delta(G)$ is complete.

In particular, simple groups have complete class-size prime graph.

As it can be perceived, non-adjacency between vertices of $\Delta(G)$ highly
restricts the group structure. Indeed, the most extreme case was analysed in
[10] by Bertram, Herzog and Mann, where they characterised the structure of
those groups $G$ whose _class-size common divisor graph_ $\Gamma(G)$ is
disconnected; the vertices of $\Gamma(G)$ are the non-central conjugacy
classes of $G$, and two classes are adjacent if their sizes are not coprime.
Certainly, $\Gamma(G)$ is connected if and only if $\Delta(G)$ is connected,
so these authors particularly provided a characterisation of those groups
whose class-size prime graph is disconnected. Later on, this fact was
independently proved by Dolfi in [20, Theorem 4] using different tools:

######  Theorem 2.6.

_([10, Theorem 2])._ Let $G$ be a finite group. Then $\Delta(G)$ is
disconnected if and only if up to abelian direct factors, $G=A\rtimes B$ with
$A$ and $B$ abelian, coprime Hall subgroups, and
$G/\boldsymbol{\operatorname{Z}}(G)$ is a Frobenius group.

In this last situation, $\Delta(G)$ turns out to be the union of two complete
connected components (also called _cliques_), which are the sets of prime
divisors of the Frobenius kernel and a Frobenius complement of
$G/\boldsymbol{\operatorname{Z}}(G)$, respectively.

In parallel to this research, Chillag and Herzog examined in [19] a more
restrictive situation, namely groups all of whose class sizes are prime
powers. That result, which can be viewed as a consequence of the above one,
can be restated in terms of the class-size prime graph as follows.

######  Corollary 2.7.

_([19, Theorem 2 and Corollary 2.2])._ Let $G$ be a finite group. Then
$\Delta(G)$ has no edges if and only if up to abelian direct factors:

  * _(a)_

$G$ is a non-abelian $p$-group, for some prime $p$; or

  * _(b)_

$G=Q\rtimes P$ with $P$ a $p$-group, $Q$ a $q$-group ($p,q$ primes), both $P$
and $Q$ abelian, and $G/\boldsymbol{\operatorname{Z}}(G)$ a Frobenius group.

Interestingly, that feature of $\Delta(G)$ in the disconnected case of having
two complete induced subgraphs is not sparse. In fact, Alfandary proved in [2]
that if $G$ is soluble and it does not contain a certain subgroup of an affine
semi-linear group, then $\Delta(G)$ also possesses two complete induced
subgraphs. Further, this situation turns out to hold in full generality for
every finite group (see Corollary 3.4).

As it will be shown in the next section (see Theorem 3.1), the diameter of
$\Delta(G)$ in the connected case is at most $3$. Casolo and Dolfi in 1996
characterised those groups for which the diameter of $\Delta(G)$ attains this
maximal bound:

######  Theorem 2.8.

_([15, Theorem 8])._ Let $G$ be a group such that $\Delta(G)$ is connected.
Then the diameter of $\Delta(G)$ is equal to $3$ if and only if $G=A\rtimes B$
where $A$ and $B$ are abelian, coprime Hall subgroups, and there exist
$p\in\pi(B)$, $q\in\pi(A)$, $B_{p}\in\operatorname{Syl}_{p}\left(B\right)$,
and $A_{q}\in\operatorname{Syl}_{q}\left(A\right)$ such that:

  * _(a)_

$1\neq\boldsymbol{\operatorname{C}}_{A}(B_{p})\neq A$,
$1\neq\boldsymbol{\operatorname{C}}_{B}(A_{q})\neq B$;

  * _(b)_

$\boldsymbol{\operatorname{C}}_{A}(y)\leqslant\boldsymbol{\operatorname{C}}_{A}(B_{p})$
for all $y\in B\smallsetminus\boldsymbol{\operatorname{Z}}(G)$;

  * _(c)_

$\boldsymbol{\operatorname{C}}_{B}(x)\leqslant\boldsymbol{\operatorname{C}}_{B}(A_{q})$
for all $x\in A\smallsetminus\boldsymbol{\operatorname{Z}}(G)$.

In particular, in [15, Proposition 7] it is also showed that if $G$ is non-
soluble, then the diameter of $\Delta(G)$ is at most $2$. This bound is best
possible, as Casolo and Dolfi noticed with the following example: take the
direct product of an alternating group on $5$ letters and a metacyclic
Frobenius group of order coprime to $30$.

The next research on this topic focused on refining the previously mentioned
Itô’s result (Theorem 2.4 (a)). That theorem explores the structure of $G$
focusing on a prime $p$ and a single prime non-adjacent to it. In this spirit,
Casolo and Dolfi considered in [15, Lemma 5] a global, rather than local,
perspective; that is, they analysed the structure of a soluble group $G$ via
the non-adjacency of $p$ and a set of vertices $\pi$ distinct from $p$, by
proving that the Hall $\pi$-subgroups of $G$ are abelian. Moreover, in the
joint work [18] of these authors with Pacifici and Sanus, the solubility
hypothesis was removed:

######  Theorem 2.9.

_([18, Theorem C])._ Let $G$ be a group, $p$ a vertex of $\Delta(G)$, and
$\pi$ a subset of vertices that are non-adjacent to $p$ and different from
$p$. Then G is $\pi$-soluble, and the following conclusions hold.

  * _(a)_

$G$ has abelian Hall $\pi$-subgroups and $\pi$-length 1.

  * _(b)_

The vertices in $\pi$ are pairwise adjacent.

In that paper, and still in the spirit of a global perspective, the authors
also studied how the set of incomplete vertices (i.e. vertices that are not
adjacent to at least one other vertex) of $\Delta(G)$ influences the group
structure:

######  Theorem 2.10.

_([18, Theorem A])._ Let $G$ be a group, and let $\pi_{0}$ be the set of
incomplete vertices of $\Delta(G)$. Then $G$ admits a Hall $\pi_{0}$-subgroup
$H$, and the following conclusions hold.

  * _(a)_

$H$ is metabelian.

  * _(b)_

$H\cap G^{\prime\prime}\leqslant\boldsymbol{\operatorname{F}}(G)$.

Certainly, statement (b) is equivalent to saying that the Hall
$\pi_{0}$-subgroups of $G^{\prime\prime}$ are nilpotent and subnormal in
$G^{\prime\prime}$, and therefore
$|G^{\prime\prime}/\boldsymbol{\operatorname{F}}(G^{\prime\prime})|$ is not
divisible by any prime in $\pi_{0}$. This fact and Burnside’s
$p^{\alpha}q^{\beta}$-theorem lead to the next result.

######  Corollary 2.11.

_([18, Corollary B])._ Let $G$ be a group. Then the following conclusions
hold.

  * _(a)_

All the primes within
$\pi(G^{\prime\prime}/\boldsymbol{\operatorname{F}}(G^{\prime\prime}))$ are
complete vertices of $\Delta(G)$.

  * _(b)_

If $\Delta(G)$ has at most two complete vertices, then $G$ is soluble.

Note that the class-size prime graph of an alternating group on 5 letters has
three complete vertices, and clearly that group is non-soluble.

In an earlier paper, Casolo _et al._ addressed a more restrictive situation,
namely $\Delta(G)$ has at most one complete vertex. Under this assumption,
they proved in [17, Theorem 2.4] not only the solubility of $G$, but also that
$G^{\prime}\boldsymbol{\operatorname{F}}(G)/\boldsymbol{\operatorname{F}}(G)$
is nilpotent, so $G$ is nilpotent-by-nilpotent-by-abelian. Further, if the
prime $2$ is not a complete vertex of $\Delta(G)$, then $G$ is in fact
nilpotent-by-metabelian, and they showed that this description is, in some
sense, best possible (see [17, Remark 2.5 and Example 2.6]).

In the most extreme case when no vertex of $\Delta(G)$ is complete, by Theorem
2.10 we get that $G$ is metabelian, and the following stronger conclusion was
obtained by the previous authors:

######  Theorem 2.12.

_([17, Theorem C])._ Let $G$ be a group, and assume that no vertex of
$\Delta(G)$ is complete. Then, up to abelian direct factors, $G=K\rtimes H$
where $K$ and $H$ are abelian, coprime Hall subgroups. Moreover,
$K=G^{\prime}$, $K\cap\boldsymbol{\operatorname{Z}}(G)=1$, and both $\pi(K)$
and $\pi(H)$ induce complete subgraphs in $\Delta(G)$.

Graphs that are non-complete and regular clearly satisfy the above property,
and in that paper a complete characterisation of groups with such class-size
prime graphs is attained. Before presenting it, we recall that the _join_ of
two graphs $\Delta_{1}$ and $\Delta_{2}$, with disjoint vertex sets $V_{1}$
and $V_{2}$, is the graph $\Delta_{1}\ast\Delta_{2}$ whose vertex set is
$V_{1}\cup V_{2}$, and two vertices are adjacent whenever either one of them
lies in $V_{1}$ and the other one in $V_{2}$, or they are vertices of the same
$\Delta_{i}$ and they are adjacent in $\Delta_{i}$ for some $i\in\\{1,2\\}$.
Besides, in that paper, the concept of a $\mathcal{D}$_-group_ is used for
groups $G=A\rtimes B$ where $A$ and $B$ are abelian, coprime Hall subgroups,
$\boldsymbol{\operatorname{Z}}(G)\leqslant B$, and
$G/\boldsymbol{\operatorname{Z}}(G)$ is a Frobenius group.

######  Theorem 2.13.

_([17, Theorem D])._ Let $G$ be a group, and assume that $\Delta(G)$ is a non-
complete regular graph of degree $d$ with $n$ vertices. Then, up to abelian
direct factors, $G=G_{1}\times\cdots\times G_{n/2m}$, where $m=(n-1)-d$, and
$G_{i}$ are pairwise coprime $\mathcal{D}$-groups satisfying that the orders
of both the Frobenius kernel and the Frobenius complements of
$G_{i}/\boldsymbol{\operatorname{Z}}(G_{i})$ have $m$ prime divisors.

Conversely, for a group $G$ of this kind, $\Delta(G)$ is the join of $n/2m$
copies of a graph having two complete connected components of $m$ vertices
each.

As a direct consequence, we deduce that $\Delta(G)$ is a square with vertex
set $V(G)=\\{p,q,r,s\\}$ and edge set
$E(G)=\\{\\{p,q\\},\\{p,r\\},\\{q,s\\},\\{r,s\\}\\}$ if and only if $G=A\times
G_{1}\times G_{2}$ where $G_{1}$ and $G_{2}$ are $\mathcal{D}$-groups with
$\pi(G_{1})=\\{p,s\\}$ and $\pi(G_{2})=\\{q,r\\}$, and $A$ is abelian.

A natural generalisation of this graph arises when one replaces each vertex by
a set of vertices. With this in mind, a (simple undirected) graph is said to
be a _block square_ if its vertex set can be partitioned into four disjoint,
non-empty subsets $\pi_{1},\pi_{2},\pi_{3},\pi_{4}$ such that no prime in
$\pi_{1}$ is adjacent to any prime in $\pi_{3}$, no prime in $\pi_{2}$ is
adjacent to any prime in $\pi_{4}$, and there exist vertices in both $\pi_{1}$
and $\pi_{3}$ that are adjacent to vertices in $\pi_{2}$ and in $\pi_{4}$.

The result below, which I proved in [40], provides a complete characterisation
of those groups whose class-size prime graph is a block square.

######  Theorem 2.14.

_([40, Theorem A])._ Let G be a finite group. Then $\Delta(G)$ is a block
square if and only if, up to an abelian direct factor, $G=A\times B$ where $A$
and $B$ are $\mathcal{D}$-groups of coprime orders.

We close this section with recent research on how non-adjacency features of
$\Delta(G)$ control the group structure. Let $\Delta$ be a graph with $n$
connected components. A vertex $v$ of $\Delta$ is said to be a _cut-vertex_ if
the number of connected components of the graph $\Delta-v$, obtained by
removing the vertex $v$ and all edges incident to $v$ from $\Delta$, is larger
than $n$.

Dolfi _et al._ carried out in [27, Theorem 3.3] a complete, and somewhat
technical, characterisation of the structure of a group $G$ such that
$\Delta(G)$ has a cut vertex. In particular, such a group $G$ turns out to be
soluble with Fitting height at most 3, and its Sylow $p$-subgroups are abelian
for every prime $p$ distinct from the cut vertex (cf. [27, Theorem A (a)]).
These authors also characterised in [27, Theorem C] groups whose class-size
prime graph has two cut vertices (which is actually the maximal number of cut
vertices that can occur, as it will be shown in the next section). As an
application, a classification of the finite groups $G$ such that $\Delta(G)$
has no cycle as an induced subgraph is achieved (cf. [27, Corollary 3.4]).

##  3 Graphs that may occur as $\Delta(G)$

In this section, we put focus on which graphs $\Delta$ might satisfy that
$\Delta(G)=\Delta$ for some finite group $G$. Recall that, in virtue of
Theorem 2.6, the number of connected components of $\Delta(G)$ is less than or
equal to $2$ for every group $G$, having both components diameter $1$. It is
natural, then, to examine whether the diameter of $\Delta(G)$ can be upper
bounded in the connected case. This problem was addressed by Alfandary in [1],
where he proved the result below.

######  Theorem 3.1.

_([1, Corollary 2.5])._ Let $G$ be a group. If $\Delta(G)$ is connected, then
its diameter is at most $3$.

Dolfi proved the same conclusion in [20, Theorem 17], and he provided an
example which shows that this bound is best possible.

Roughly speaking, the class-size prime graph is very rich in edges. This
property can be deduced from the next theorem, whose proof involves the CFSG
via Theorem 2.5.

######  Theorem 3.2.

_([21, Theorem A])._ Let $G$ be a finite group. If $p,q,r\in V(G)$ are
pairwise distinct, then two among them are adjacent in $\Delta(G)$.

It is worthwhile to point out that this fact was firstly proved for soluble
groups by Dolfi itself in [20]. From the above result, it follows again the
bound of $3$ for the diameter of $\Delta(G)$ in the connected case, and also
the feature of $\Delta(G)$ in the non-connected case of being the union of two
complete subgraphs.

At this point, it is convenient to recall the definition of the complement
graph $\overline{\Delta(G)}$ of the class-size prime graph: this graph has the
same vertex set $V(G)$, but two vertices are adjacent in
$\overline{\Delta(G)}$ if and only if they are not adjacent in $\Delta(G)$.
With this in mind, Theorem 3.2 can be expressed as follows: the graph
$\overline{\Delta(G)}$ does not contain any cycle of length $3$, for any group
$G$. Now it is natural to wonder whether there exist cycles of larger lengths
within $\overline{\Delta(G)}$; and indeed any Frobenius group $G$ with abelian
kernel and complement, both with orders divisible by two different primes,
provides a cycle of length $4$ in $\overline{\Delta(G)}$. However, Dolfi _et
al._ showed in [26] that it is not possible to find cycles of length $5$, and
in general of any odd length:

######  Theorem 3.3.

_([26, Theorem A])._ For any group $G$, the graph $\overline{\Delta(G)}$ does
not contain any cycle of odd length.

A first immediate consequence is that the class-size prime graph of a group
cannot contain any pentagon as an induced subgraph. Further, the assertion of
the above theorem is equivalent to the fact that $\overline{\Delta(G)}$ is
always a bipartite graph, and so the following result directly follows.

######  Corollary 3.4.

_([26, Corollaries B and C])._ For every finite group $G$, the vertex set of
$\Delta(G)$ can be partitioned in two subsets of pairwise adjacent vertices.
In particular, if $\omega(G)$ is the maximum size of a clique in $\Delta(G)$,
then it holds $|V(G)|\leq 2\omega(G)$.

Some results that have appeared in the previous section lead to
characterisations of which non-complete regular graphs and which block square
graphs may occur as $\Delta(G)$ for some group $G$. For instance, the
following two results are consequences of Theorems 2.13 and 2.14,
respectively.

######  Theorem 3.5.

_([17, Corollary E])._ Let $\Delta$ be a non-complete regular graph. Then
there exists a finite group $G$ such that $\Delta(G)=\Delta$ if and only if
$\Delta$ is the join of $k$ copies of a graph that have two complete connected
components of $m$ vertices each, for some positive integers $k$ and $m$.

######  Theorem 3.6.

_([40, Corollary B])._ Let $\Delta$ be a block square graph. Then there exists
a finite group $G$ such that $\Delta(G)=\Delta$ if and only if $\pi_{i}$ is a
clique for each $1\leq i\leq 4$, and all the primes in $\pi_{1}\cup\pi_{4}$
are adjacent to all the primes in $\pi_{2}\cup\pi_{3}$.

In particular, it cannot occur a _house graph_ (i.e. a square graph in which
two adjacent vertices are adjacent to an additional vertex) as $\Delta(G)$ for
any group $G$.

Regarding graphs that possess a cut vertex, Dolfi _et al._ obtained the next
conclusions.

######  Theorem 3.7.

_([27, Theorem A])._ Let $G$ be a group such that $\Delta(G)$ has a cut vertex
$r$. Then the following conclusions hold:

  * _(a)_

$\Delta(G)-r$ is a graph with two connected components, that are both complete
graphs.

  * _(b)_

If $r$ is a complete vertex of $\Delta(G)$, then it is the unique complete
vertex and the unique cut vertex of $\Delta(G)$. If $r$ is non-complete, then
$\Delta(G)$ is a graph of diameter 3, and it can have at most two cut
vertices.

Hence, $\Delta(G)$ can have at most two cut vertices for any group $G$, and a
characterisation of the associated class-size prime graphs was also achieved
in [27].

######  Theorem 3.8.

_([27, Theorem D])._ Let $\Delta$ be a graph having a cut vertex. Then there
exists a finite group $G$ such that $\Delta(G)=\Delta$ if and only if $\Delta$
is connected and its vertex set can be partitioned in two subsets of pairwise
adjacent vertices.

##  4 Variations on the class-size prime graph

Over the last years, several authors have investigated whether the entire data
contained in $cs(G)$ is required for studying the structure of $G$. In
particular, some smaller subsets of elements of the group $G$ have been
considered, and hence it naturally arised the definition and the study of the
prime graphs built on the associated conjugacy class sizes. It is worthwhile
to remark that, in general, these graphs are not induced subgraphs of
$\Delta(G)$.

###  $p$-regular and $p$-singular conjugacy classes

In 1972, Camina generalised Lemma 2.3 by considering only $p$_-regular_
elements for a fixed prime $p$, i.e. elements of order not divisible by $p$.
He proved that $p$ does not divide any conjugacy class size of each
$p$-regular element of a group $G$ if and only if $G$ has a Sylow $p$-subgroup
as a direct factor. Let $cs_{p^{\prime}}(G)$ be the set of conjugacy class
sizes of $p$-regular elements of a group $G$. Then this result can be
rephrased in terms of $\Delta(cs_{p^{\prime}}(G))$:

######  Lemma 4.1.

_([13, Lemma 1], rephrased)._ Let $G$ be a group, $p$ a prime, and
$P\in\operatorname{Syl}_{p}\left(G\right)$. Then $p$ is not a vertex of
$\Delta(cs_{p^{\prime}}(G))$ if and only if $G=P\times H$ with
$H\in\operatorname{Hall}_{p^{\prime}}\left(G\right)$.

Lu and Zhang studied in [36] the number of connected components of
$\Delta(cs_{p^{\prime}}(G))$, but assuming the $p$-solubility of the group
$G$. Beltrán and Felipe attained best possible bounds for the diameter in [4]
and [5].

######  Theorem 4.2.

Let $G$ be a $p$-soluble group, for a fixed prime $p$. Then:

  * _(a)_

_([36, Theorem 1])._ The number of connected components of
$\Delta(cs_{p^{\prime}}(G))$ is at most $2$.

  * _(b)_

_([4, Theorem 4] and [5, Theorem A])._ If $\Delta(cs_{p^{\prime}}(G))$ is
connected, then its diameter is at most $3$; and if it is disconnected, then
both connected components are complete subgraphs.

Observe that these bounds cannot be directly deduced from the fact that
$\Delta(cs_{p^{\prime}}(G))$ is a subgraph of $\Delta(G)$.

In [6], Beltrán and Felipe extended this study by analysing the structure of a
$p$-soluble group $G$ with disconnected $\Delta(cs_{p^{\prime}}(G))$. The
authors did not achieved a complete characterisation, but they provided some
necessary conditions, as well as some sufficient ones.

A characterisation of $p$-soluble groups such that all $p$-regular conjugacy
classes have prime power size was addressed in [7] by the same authors:

######  Theorem 4.3.

_([7, Theorem D], rephrased)._ Let $G$ be a $p$-soluble group, for a fixed
prime $p$. Then $\Delta(cs_{p^{\prime}}(G))$ has no edges (i.e. is an empty
graph) if and only if one of the following conclusions hold:

  * _(a)_

$G$ has abelian $p$-complements. This occurs if and only if
$cs_{p^{\prime}}(G)$ only contains powers of $p$.

  * _(b)_

$G$ is nilpotent with abelian Sylow $r$-subgroups for all primes
$r\notin\\{p,q\\}$, for some prime $q\neq p$. This happens if and only if
$cs_{p^{\prime}}(G)$ only contains powers of $q$.

  * _(c)_

$G=P\times H$ with $P\in\operatorname{Syl}_{p}\left(G\right)$ and
$H\in\operatorname{Hall}_{p^{\prime}}\left(G\right)$. Furthermore, $H=A\rtimes
B$ with both $A$ and $B$ abelian, $H/\boldsymbol{\operatorname{Z}}(H)$ is a
Frobenius group, and the class sizes of $H$ are powers of either $q$ or $r$,
for some primes $q$ and $r$ distinct from $p$. This happens if and only if
$cs_{p^{\prime}}(G)$ consists of powers of $q$ and $r$.

In particular, $\Delta(cs_{p^{\prime}}(G))$ is an isolated vertex if and only
if either (a) or (b) above holds. These authors extended the previous
necessary condition in [8, Theorem A], by removing the $p$-solubility
assumption, and considering prime power class lengths of $\pi$-elements, for a
set of primes $\pi$.

Alternatively, Qian and Wang focused on the class sizes of _$p$ -singular
elements_, which are elements whose order is divisible by $p$. Let $cs_{p}(G)$
denote the set of class sizes of such elements in a group $G$. Note that if
$p\in\pi(\boldsymbol{\operatorname{Z}}(G))$, then $cs(G)=cs_{p}(G)$. In [38]
they proved the next results.

######  Theorem 4.4.

Let $G$ be a group, $p$ a prime, and
$P\in\operatorname{Syl}_{p}\left(G\right)$.

  * _(a)_

_([38, Theorem B], rephrased)._ $p$ is not a vertex of $\Delta(cs_{p}(G))$ if
and only if $P$ is abelian, $P\cap P^{g}=1$ for each $g\in
G\smallsetminus\boldsymbol{\operatorname{N}}_{G}(P)$, and
$\boldsymbol{\operatorname{N}}_{G}(P)/\boldsymbol{\operatorname{C}}_{G}(P)$
acts Frobeniusly on $P$ whenever
$\boldsymbol{\operatorname{C}}_{G}(P)<\boldsymbol{\operatorname{N}}_{G}(P)$.

  * _(b)_

_([38, Theorem C])._ If $p\notin\pi(\boldsymbol{\operatorname{Z}}(G))$, then
$\Delta(cs_{p}(G))$ is connected with diameter at most $3$.

### Real conjugacy classes

Various authors have investigated the sizes of the so-called _real conjugacy
classes_ of $G$, i.e. conjugacy classes $x^{G}$ satisfying that $\chi(x)$ is a
real number for each $\chi\in{\operatorname{Irr}}(G)$. This is analogous to
saying that $x^{g}=x^{-1}$ for some $g\in G$, so $x^{G}=(x^{-1})^{G}$. Let us
denote by $cs_{\mathbb{R}}(G)$ the set of sizes of the real conjugacy classes
of $G$.

In 2009, Navarro, Sanus and Tiep studied the situation where all the real
class sizes of $G$ are $2$-powers, which means that the set of vertices of
$\Delta(cs_{\mathbb{R}}(G))$ is just the prime $2$. In virtue of [37, Theorem
C (a)], this happens if and only if $G$ has a normal $2$-complement $K$ and
all the real elements of $G$ lie in $\boldsymbol{\operatorname{C}}_{G}(K)$.
One year before, Dolfi, Navarro and Tiep characterised when
$cs_{\mathbb{R}}(G)$ satisfies the “opposite” situation, namely all real class
sizes are odd:

######  Theorem 4.5.

_([22, Theorem 6.1], rephrased)._ Let $G$ be a group and let
$P\in\operatorname{Syl}_{2}\left(G\right)$. Then $2$ is not a vertex of
$\Delta(cs_{\mathbb{R}}(G))$ if and only if $P$ is normal in $G$ and each real
element of $P$ lies in $\boldsymbol{\operatorname{Z}}(P)$.

For odd primes, however, the approach is quite more difficult and it depends
heavily on the CFSG, which contrasts with Lemma 2.3. Combining results of
[32], [33] and [42], the next theorem follows. Recall that
$\boldsymbol{\operatorname{O}}^{p^{\prime}}(G)$ is the smaller normal subgroup
of $G$ such that the corresponding quotient group has order not divisible by
$p$.

######  Theorem 4.6.

_([43, Lemma 2.7], rephrased)._ Let $G$ be a group and $p$ be an odd prime. If
$p=3$, assume in addition that $G$ has no composition factor isomorphic to
$SL_{3}(2)$. If $p$ is not a vertex of $\Delta(cs_{\mathbb{R}}(G))$, then $G$
is $p$-soluble and $\boldsymbol{\operatorname{O}}^{p^{\prime}}(G)$ is soluble.
Furthermore, $\boldsymbol{\operatorname{O}}^{2^{\prime}}(G)$ has a normal
Sylow $p$-subgroup $P$ and
$P^{\prime}\leqslant\boldsymbol{\operatorname{Z}}(\boldsymbol{\operatorname{O}}^{2^{\prime}}(G))$.

It is also proved in [22, Theorem 6.2] that the number of connected components
of $\Delta(cs_{\mathbb{R}}(G))$ is at most $2$. In the extreme case when
$\Delta(cs_{\mathbb{R}}(G))$ is disconnected, Tong-Viet obtained the following
result.

######  Theorem 4.7.

_([43, Theorems A and B])._ Let $G$ be a group such that
$\Delta(cs_{\mathbb{R}}(G))$ is disconnected. Then $G$ is soluble, $2$ is a
vertex of $\Delta(cs_{\mathbb{R}}(G))$, and one of the following conclusions
holds.

  * _(a)_

$G$ has a normal Sylow $2$-subgroup.

  * _(b)_

$\Delta(cs_{\mathbb{R}}(\boldsymbol{\operatorname{O}}^{2^{\prime}}(G)))$ is
disconnected and
$cs_{\mathbb{R}}(\boldsymbol{\operatorname{O}}^{2^{\prime}}(G))$ are either
odd numbers or powers of $2$.

Actually, in the above statement (b), both connected components are complete
and one of them is just the prime $2$ (see [43, Theorem 3.5]). Concerning
statement (a), Tong-Viet conjectured that it cannot occur; in other words,
$\Delta(cs_{\mathbb{R}}(G))$ is connected when $G$ has a normal Sylow
$2$-subgroup. In fact, he proved that conjecture whenever the real elements of
the Sylow $2$-subgroup $P$ of $G$ are central in $P$ (see [43, Lemma 3.6]).

Regarding the solubility assertion of Theorem 4.7, the same conclusion was
achieved in [23, Theorem 3.1] by assuming a more restrictive hypothesis,
namely all real class sizes of $G$ are either odd or powers of $2$. Thus,
every group whose real classes all have prime power size is soluble, and its
structure seems to be characterised in a current preprint due to Bonazzi (see
[12]). On the other hand, the groups whose graph $\Delta(cs_{\mathbb{R}}(G))$
consists of only one odd vertex have not yet been characterised.

### Vanishing conjugacy classes

In parallel to the aforementioned research on real class sizes, some authors
started to filter the class sizes of $G$ by the set ${\operatorname{Irr}}(G)$
in a different manner. Concretely, an element $g\in G$ is said to be
_vanishing in_ $G$ if $\chi(g)=0$ for some $\chi\in{\operatorname{Irr}}(G)$,
and so $g^{G}$ is called a _vanishing conjugacy class_. This definition is
motivated by a celebrated theorem due to Burnside, which asserts that a non-
linear irreducible character always vanishes on some conjugacy class; so it is
natural to analyse the columns of the character table of a group that verify
that property. We refer the reader to the expository paper [25] for a
collection of results and conjectures on this topic.

Let $cs_{v}(G)$ be the set of vanishing class sizes of $G$. An immediate
consequence of the mentioned Burnside’s result is the next one: if
$\Delta(cs_{v}(G))$ is a null graph (i.e. it has no vertices), then $G$ is
abelian. Note that the converse also holds, so the commutativity of $G$ can be
characterised in terms of $\Delta(cs_{v}(G))$.

In [24, Theorem A], the authors studied (via the CFSG) the situation when a
given prime $p$ does not divide any vanishing conjugacy class size of a group,
which has the following transcription in terms of $\Delta(cs_{v}(G))$.

######  Theorem 4.8.

_([24, Theorem A], rephrased)._ Let $G$ be a group, and $p$ a prime. If $p$ is
not a vertex of $\Delta(cs_{v}(G))$, then $G$ has a normal $p$-complement and
abelian Sylow $p$-subgroups.

It is significant to mention that the same assertion remains true if we
consider only the class sizes of elements that are zeros of some (ordinary)
irreducible character of $G$ lying in the _principal $p$-block_, i.e.
characters $\chi\in{\operatorname{Irr}}(G)$ such that the sum of $\chi(g)$,
for all $p$-regular elements $g\in G$, is non-zero (cf. [25, Theorem 6.3]).

Certainly, the vertex set of $\Delta(cs_{v}(G))$ can be smaller than that of
$\Delta(G)$: it is enough to consider a symmetric group on $3$ letters.
Surprisingly enough, if $G$ possesses a non-abelian minimal normal subgroup,
then Bianchi _et al._ proved that these two vertex sets turn out to be equal:

######  Theorem 4.9.

_([11, Proposition])._ Let $G$ be a group that has a non-abelian minimal
normal subgroup. Then the vertex sets of $\Delta(G)$ and $\Delta(cs_{v}(G))$
are equal.

In the spirit of [16], and under the assumption that $G$ has a non-abelian
minimal normal subgroup, these authors also obtained the next “vanishing
versions” of Theorems 2.4 (b) and Theorem 2.5:

######  Theorem 4.10.

_([11, Theorem A])._ Let $G$ be a group having a non-abelian minimal normal
subgroup. If $\\{p,q\\}$ are vertices of $\Delta(cs_{v}(G))$ that are non-
adjacent, then $G$ is $\\{p,q\\}$-soluble.

######  Theorem 4.11.

_([11, Theorem B])._ Let $G$ be a group with
$\boldsymbol{\operatorname{F}}(G)=1$. Then $\Delta(cs_{v}(G))$ is a complete
graph with vertex set $\pi(G)$.

An immediate $p$-solubility criterion follows from Theorems 4.8 and 4.10: if
$p$ is not a complete vertex of $\Delta(cs_{v}(G))$, and $G$ has a non-abelian
minimal normal subgroup, then $G$ is $p$-soluble (cf. [11, Corollary]).
Besides, as noticed in [11], the assumption about the existence of a non-
abelian minimal normal subgroup in Theorem 4.10 is fundamental.

All the previous results might suggest that, under the assumption that $G$ has
a non-abelian minimal normal subgroup, the graphs $\Delta(G)$ and
$\Delta(cs_{v}(G))$ actually coincide, which is an open question.

In 2019, Felipe, Martínez-Pastor and myself analysed the influence of class
sizes of vanishing elements of prime power order on the structure of certain
factorised groups. In particular, from [31, Corollary 4] it can be obtained
some information about groups satisfying that the vanishing class sizes are
all prime powers, i.e. groups $G$ such that $\Delta(cs_{v}(G))$ is an empty
graph.

######  Corollary 4.12.

Let $G$ be a group. If $\Delta(cs_{v}(G))$ has no edges, then
$G/\boldsymbol{\operatorname{F}}(G)$ is abelian.

This result has also been proved independently by Robati and Hafezieh-Balaman
in [39, Corollary 2.3]. In that paper, it is also addressed the particular
situation in which the lenghts of the vanishing conjugacy classes of a group
are all powers of a fixed prime $p$; in other words, it is addressed the
feature of $\Delta(cs_{v}(G))$ of being a single vertex $p$ (see [39, Theorem
1.1]).

Recently, the authors of [28] have further filtered the set of class sizes of
$G$ by considering only the _SM-vanishing conjugacy classes_ , i.e. conjugacy
classes that are zeros of the _strongly monolithic_ characters of $G$. It is
worth mentioning that some of the above situations also hold for the
corresponding prime graph built on this last subset of class sizes of $G$ (cf.
[28]).

### Normal subgroups

Beltrán, Felipe and Melchor considered in [9] the set $cs_{G}(N)$ of conjugacy
class sizes in $G$ of elements of a normal subgroup $N$ of $G$, and they
introduced the associated prime graph $\Delta(cs_{G}(N))$. In that paper, they
obtained the next result.

######  Theorem 4.13.

_([9, Theorems C and D])._ Let $N$ be a normal subgroup of a group $G$. Then
the number of connected components of $\Delta(cs_{G}(N))$ is at most $2$.
Moreover, if $\Delta(cs_{G}(N))$ is connected, then its diameter is at most
$3$; and if $\Delta(cs_{G}(N))$ is disconnected, then each connected component
is a complete graph.

Additionally, in the disconnected case, the authors also obtained strong
restrictions on the structure of the normal subgroup $N$:

######  Theorem 4.14.

_([9, Theorem E])._ Let $N$ be a normal subgroup of a group $G$. If
$\Delta(cs_{G}(N))$ is disconnected, then either $N=P\times A$ with $P$ a
$p$-group and $A\leqslant\boldsymbol{\operatorname{Z}}(G)$, or $N=K\rtimes H$
where both $K$ and $H$ are abelian subgroups and
$N/\boldsymbol{\operatorname{Z}}(N)$ a Frobenius group.

Note that in the previous result it appears a structure case for $N$ that
differs with the general case of $N=G$ in Theorem 2.6.

### Products of groups

Another variation arises when one consider a factorised group and the
conjugacy class sizes of only the elements that lie in the factors. More
concretely, if $G=AB$ is the product of two subgroups $A$ and $B$, then we can
define $cs_{G}(A\cup B)$ as the set of class sizes of the elements that belong
to $A\cup B$. Felipe _et al._ proved in [29, Corollary 1] that, in a
factorised group $G=AB$, all the prime power order elements in $A\cup B$ have
class sizes not divisible by a fixed prime $p$ if and only if $G$ has a
central Sylow $p$-subgroup. Therefore, the next consequence directly follows.

######  Corollary 4.15.

Let the group $G=AB$ be the product of subgroups $A$ and $B$, and let $p$ be a
prime. Then $p$ is not a vertex of $\Delta(cs_{G}(A\cup B))$ if and only if
$G$ has a central Sylow $p$-subgroup.

The proof of [29, Corollary 1] uses the CFSG, in contrast to the elementary
proof of Lemma 2.3.

In [30], Felipe, Martínez-Pastor and myself introduced the concept of _Baer
factorisations_ $G=AB$, that is, products of groups such that the class sizes
in $G$ of the prime power order elements in $A\cup B$ are also prime powers.
This is a generalisation of the so-called _Baer groups_ (see [3]). In [30,
Corollary 5], we characterised the structure of $G$ in the particular
situation when those class sizes are precisely powers of a fixed prime $p$. In
particular, this holds when $cs_{G}(A\cup B)$ only contains powers of $p$, and
so the next result follows.

######  Corollary 4.16.

Let the group $G=AB$ be the product of subgroups $A$ and $B$. Then
$\Delta(cs_{G}(A\cup B))$ is an isolated vertex $p$ if and only if $G$ has a
central $p$-complement.

More generally, we obtained some properties that are verified by Baer
factorisations, and thus we can derive the same conclusions for a factorised
group $G=AB$ such that $\Delta(cs_{G}(A\cup B))$ has no edges. For instance,
we proved that $G/\boldsymbol{\operatorname{F}}(G)$ is abelian, and both $A$
and $B$ are Baer groups (cf. [30]). Indeed, using [30, Proposition D], it can
be verified that if $\Delta(cs_{G}(A\cup B))$ has no edges, then both
$\Delta(A)$ and $\Delta(B)$ have no edges (although in general they are not
subgraphs of $\Delta(G)$), and therefore by Corollary 2.7 we can control the
structure of both subgroups.

As demonstrated in the previous subsections, if some information on $cs(G)$ is
not considered, then it is not always possible to get analogous results to the
ordinary case. As an example, Itô’s result (Theorem 2.4 (a)) states that if
$\\{p,q\\}$ are non-adjacent vertices of $\Delta(G)$, then $G$ has either a
normal $p$-complement or a normal $q$-complement. It is natural to wonder
whether an analogous statement can be obtained when we consider this property
on $\Delta(cs_{G}(A\cup B))$. Nevertheless, if $G=A\times B$ is the direct
product of a symmetric group $A$ on $3$ letters and a dihedral group $B$ of
order $10$, then $cs_{G}(A\cup B)=\\{2,3,5\\}$; so $3$ and $5$ are non-
adjacent vertices of $\Delta(cs_{G}(A\cup B))$, and $G$ has neither a normal
$3$-complement nor a normal $5$-complement. But $G$ has $\\{3,5\\}$-length
$1$, and both the Sylow $3$-subgroup and $5$-subgroup are abelian, so it is an
open question whether the claims in Theorem 2.4 (b) hold for factorised groups
too.

Regarding the number of connected components of $\Delta(cs_{G}(A\cup B))$,
following the techniques in [9] it is not difficult to prove that it can be at
most $4$. But the structure of $A$ and $B$ (and possibly that of $G$) in the
disconnected case, and the diameter of $\Delta(cs_{G}(A\cup B))$ in the
connected case are problems not addressed at the moment.

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