# A survey on conjugacy class graphs of groups

Peter J. Cameron, Firdous Ee Jannat, Rajat Kanti Nath and Reza Sharafdini  P.
J. Cameron, School of Mathematics and Statistics, University of St Andrews,
North Haugh, St Andrews, Fife, KY16 9SS, UK<EMAIL_ADDRESS>Firdous
Ee Jannat, Department of Mathematical Science, Tezpur University, Napaam
-784028, Sonitpur, Assam, India<EMAIL_ADDRESS>Rajat Kanti Nath,
Department of Mathematical Science, Tezpur University, Napaam -784028,
Sonitpur, Assam, India<EMAIL_ADDRESS>Reza Sharafdini, Department
of Mathematics, Persian Gulf University, Bushehr 75169-13817, Iran.
<EMAIL_ADDRESS>
###### Abstract.

There are several graphs defined on groups. Among them we consider graphs
whose vertex set consists conjugacy classes of a group $G$ and adjacency is
defined by properties of the elements of conjugacy classes. In particular, we
consider commuting/nilpotent/solvable conjugacy class graph of $G$ where two
distinct conjugacy classes $a^{G}$ and $b^{G}$ are adjacent if there exist
some elements $x\in a^{G}$ and $y\in b^{G}$ such that $\langle x,y\rangle$ is
abelian/nilpotent/solvable. After a section of introductory results and
examples, we discuss all the available results on connectedness, graph
realization, genus, various spectra and energies of certain induced subgraphs
of these graphs. Proofs of the results are not included. However, many open
problems for further investigation are stated.

###### Key words and phrases:

Commuting/Nilpotent/Solvable conjugacy class graph, connectedness, genus,
spectrum and energy, induced subgraph.

###### 2020 Mathematics Subject Classification:

Primary: 20D60, 20E45; Secondary: 05C25.

##  1\. Introduction

Characterizing finite groups using graphs defined over groups has gained
traction as a research topic in recent times. A number of graphs have been
defined on groups (see [22]), among which the commuting graph has been studied
widely. Let $G$ be a finite non-abelian group. The _commuting graph_ of $G$ is
a simple undirected graph whose vertex set is $G$, in which two vertices $x$
and $y$ are adjacent if they commute. The complement of this graph is the
_non-commuting graph_ of $G$. The concept of commuting graph appeared in an
important work of Brauer and Fowler [20], in the year 1955, a step towards the
Classification of Finite Simple Groups. After the work of Erdős and Neumann
[62] on its complement in the year 1976, it was studied in its own right.

The property that $x$ and $y$ commute is equivalent to saying that $\langle
x,y\rangle$ is abelian. Using other group types such as cyclic, nilpotent,
solvable, … graphs have been defined on groups. Given a group type
$\mathcal{P}$ (for instance, cyclic, abelian, nilpotent, solvable etc.), we
define a graph on a group $G$, called the $\mathcal{P}$ graph of $G$, whose
vertex set is $G$ and two distinct vertices $x$ and $y$ are adjacent if
$\langle x,y\rangle$ is a $\mathcal{P}$ group. In this nomenclature ‘abelian
graph’ is nothing but the commuting graph. These graphs forms the following
hierarchy (where $A\subseteq B$ denotes that $A$ is a spanning subgraph of
$B$):

(1) Cyclic graph $\subseteq$ Commuting graph $\subseteq$ Nilpotent graph
$\subseteq$ Solvable graph.

It is worth mentioning that there are other graphs in the above hierarchy (for
details one can see [22]).

A _dominant vertex_ of a graph is a vertex that is adjacent to all other
vertices. Let $\mathcal{P}(G)=\\{g\in G:\langle g,h\rangle\text{ is a
$\mathcal{P}$ group for all }h\in G\\}$. Then $\mathcal{P}(G)$ is the set of
all dominant vertices of $\mathcal{P}$ graph of $G$.

In the four cases just described, $\mathcal{P}(G)$ is a subgroup of $G$, the
cyclicizer, centre, hypercentre, and solvable radical of $G$ respectively. The
question of connectedness of the subgraphs of $\mathcal{P}$ graph induced by
$G\setminus\mathcal{P}(G)$ is an interesting problem (see [49, 38, 37, 58, 18,
7, 21]). Of course, this problem is trivial unless $\mathcal{P}(G)$ is removed
(otherwise the graph has diameter at most $2$). However for other studies such
as independence number or clique number, it makes either no difference or just
a trivial difference.

Graphs are also defined from (finite) groups by considering the vertex set as
the set of conjugacy classes (or class sizes), with adjacency defined by
certain properties of the elements of conjugacy classes or the class sizes. A
survey on graphs whose vertex set consists of class sizes of a finite group
can be found in [54]. Graphs whose vertex set consists conjugacy classes of a
group and adjacency is defined by properties of their sizes were first
considered in [12].

In this survey, we shall consider graphs whose vertex set consists conjugacy
classes of a group $G$, with adjacency defined by properties of the elements
of these classes. We call such a graph the _$\mathcal{P}$ conjugacy class
graph_ of $G$, or for short the $\mathcal{P}$CC-graph. The $\mathcal{P}$
conjugacy class graph of $G$ is a simple undirected graph whose vertex set is
the set of all the conjugacy classes of $G$ and two vertices (conjugacy
classes) $a^{G}$ and $b^{G}$ are adjacent if there exist some elements $x\in
a^{G}$ and $y\in b^{G}$ such that $\langle x,y\rangle$ is a $\mathcal{P}$
group. We have the following hierarchy in case of $\mathcal{P}$ conjugacy
class graph:

(2) Cyclic CC-graph $\subseteq$ CCC-graph $\subseteq$ NCC-graph $\subseteq$
SCC-graph,

where here and subsequently we use CCC-graph, NCC-graph, SCC-graph to denote
commuting, nilpotent and solvable conjugacy class graph. (Commuting conjugacy
class graph is synonymous with ‘abelian conjugacy class graph’.) Clearly,
$1^{G}$ (the conjugacy class of the identity element) is a dominant vertex if
$\mathcal{P}(G)$ is a subgroup. To make the question of connectedness
interesting, we should consider the induced subgraph on the set of non-
dominant vertices. However, as we will see, it is not always known what this
set is. Sometimes we just remove the identity class from the vertex set.

Note that the cyclic conjugacy class graph of a group has not yet been
studied. In what follows, we shall consider commuting/nilpotent/solvable
conjugacy class graph.

An outline of the paper follows. In the next section we give some general
results about conjugacy class graphs, including a discussion of when they are
complete (Theorem 2.3), when it happens that the $\mathcal{P}$ graph is a
“blow-up” of the $\mathcal{P}CC$-graph (Theorem 2.6), and some discussion of
the dominant vertices (a characterisation is known only for the CCC-graph, see
Proposition 2.5 and Problem 2.2). The following three sections survey the
three graph types, discussing connectedness, detailed structure for special
groups, and properties such as genus, spectrum and energy. (These results are
taken from the literature and proofs are not given.) The final section
includes some open problems.

##  2\. General remarks and examples

We begin with a general observation about conjugacy class graphs which is
often useful. Let $\mathcal{P}$ be any group-theoretic property, and let
$\Gamma$ be the $\mathcal{P}CC$-graph of $G$; that is, the vertices are
conjugacy classes, and there is an edge $\\{C_{1},C_{2}\\}$ if and only if
there exist $g_{i}\in C_{i}$ for $i=1,2$ such that the group $\langle
g_{1},g_{2}\rangle$ has property $\mathcal{P}$. In fact a stronger condition
holds: if $\\{C_{1},C_{2}\\}$ is an edge, then for any $h_{1}\in C_{1}$ there
exists $h_{2}\in C_{2}$ such that $\langle h_{1},h_{2}\rangle$ has property
$\mathcal{P}$. For let $g_{1},g_{2}$ be as in the definition. There exists
$x\in G$ such that $g_{1}^{x}=h_{1}$; then, letting $h_{2}=g_{2}^{x}$, we see
that

$\langle g_{1},g_{2}\rangle^{x}=\langle h_{1},h_{2}\rangle,$

and since $\mathcal{P}$ is a group-theoretic property it is preserved by
conjugation.

The first result relevant to conjugacy class graphs is the theorem of Landau
[53] from 1903: given the number of conjugacy classes of a finite group $G$,
there is an upper bound on the order of $G$. This implies the following
result.

######  Proposition 2.1.

Given a graph $\Gamma$, there are only finitely many finite groups $G$ whose
commuting, nilpotent or solvable conjugacy class graph is isomorphic to
$\Gamma$.

The number of such groups is not usually $1$. For example, for any abelian
group of order $n$, the CCC-graph is the complete graph on $n$ vertices.

######  Problem 2.1.

Which finite groups are uniquely determined by their commuting, nilpotent, or
solvable conjugacy class graph?

$\scriptstyle{1111}$$\scriptstyle{13}$$\scriptstyle{22}$$\scriptstyle{4}$$\scriptstyle{112}$
Figure 1. Conjugacy class graphs of $S_{4}$

Figure 1 shows the conjugacy classes of the symmetric group $S_{4}$, with the
CCC- and NCC-graphs. (The conjugacy classes of $S_{n}$ are defined by cycle
types, and are labelled with partitions of $n$; so their number is $p(n)$,
where $p$ is the partition function, sequence A000041 in the On-line
Encyclopedia of Integer Sequences [64].) Solid lines show edges in the CCC-
graph, while the dotted line is the additional edge in the NCC-graph. Note
that the partitions $1111$, $112$, $22$ and $4$ form a clique in the NCC-
graph. This observation leads to our first general result.

######  Proposition 2.2.

Let $G$ be a finite group and $p$ a prime. Then the conjugacy classes of
elements of $p$-power order form a clique in the NCC-graph.

###### Proof.

Let $P$ be a Sylow $p$-subgroup of $G$. By Sylow’s theorem, every element of
$p$-power order is conjugate to an element of $P$; so $P$ meets every
conjugacy class of $p$-elements. Let $C_{1}$ and $C_{2}$ be two such classes,
and take $g_{i}\in C_{i}\cap P$. Then $\langle g_{1},g_{2}\rangle\leq P$, so
this group is a $p$-group, hence nilpotent. ∎

Thus the NCC-graph of $S_{n}$ has a clique whose size is the number of
partitions of $n$ into powers of $2$ (sequence A018819 in the OES).

The SCC-graph of $S_{4}$ is complete, because $S_{4}$ is a solvable group (and
the class of solvable groups is subgroup-closed). This is also a special case
of the following general result.

######  Theorem 2.3.

Let $G$ be a finite group. Then the CCC-graph (resp., the NCC-graph, the SCC-
graph) of $G$ is complete if and only if $G$ is abelian (resp., nilpotent,
solvable).

The “if” statements are clear. For the converses, we use the following result.
Its roots lie in the work of Jordan.

######  Proposition 2.4.

Let $H$ be a proper subgroup of the finite group $G$. Then $G$ has a conjugacy
class disjoint from $H$.

This holds because $G$ acts transitively on the set of right cosets of $H$ by
right multiplication; by Jordan’s theorem, $G$ contains an element $x$ fixing
no coset, that is, no conjugate of $x$ lies in $H$.

Thus, if we choose one element from each conjugacy class of $G$, these
elements generate $G$.

Now we prove Theorem 2.3 for the CCC-graph. Suppose that the CCC-graph is
complete. Choose any element $h$, say $h\in C_{1}$. By our general remark
about conjugacy class graphs, there exist $h_{i}\in C_{i}$ for all $i$ such
that $h_{i}$ commutes with $h=h_{1}$. Then $\langle
h_{1},\ldots,h_{r}\rangle=G$, and so $h\in Z(G)$. Since $h$ was arbitrary, $G$
is abelian.

For the SCC-graph, this is immediate from the main theorem in [24], according
to which $G$ is solvable if and only if, for all $g,h\in G$, there exists
$x\in G$ such that $\langle g,h^{x}\rangle$ is solvable.

For the NCC-graph, we use [24, Corollary E], which states that a finite group
$G$ is nilpotent if the following condition holds: for distinct primes $p,q$,
and for $g,h\in G$ where $g$ is a $p$-element and $h$ a $q$-element, there
exists $x\in G$ such that $g$ commutes with $h^{x}$. Now suppose that $G$ has
complete NCC-graph, and let $p,q,g,h$ be as stated. Then there exists $x\in G$
such that $\langle g,h^{x}\rangle$ is nilpotent. But in a nilpotent group, a
$p$-element and a $q$-element necessarily commute.

We would like to have a strengthening of this describing the dominant vertices
of one of our graphs (those joined to all others). The dominant vertices of
the commuting, nilpotent and solvable graphs are known; they are respectively
the centre, hypercentre, and solvable radical of the group [22, Theorem 11.2].
One might expect that the analogous result would hold for conjugacy class
graphs, since each of these sets is the union of conjugacy classes. But this
is not the case. The groups $\mathrm{PSL}(2,2^{a})$ for $a\geq 2$ have a
single conjugacy class of involutions, and every element is conjugate to its
inverse by some involution. This means that for any element $g$ there is an
involution $h$ such that $\langle g,h\rangle$ is dihedral. So the class of
involutions is dominant in the SCC-graph, even though the solvable radical is
trivial. It is, however, true for the CCC-graph:

######  Proposition 2.5.

The set of dominant vertices in the CCC-graph of a finite group $G$ is the set
of central conjugacy classes of $G$.

###### Proof.

Clearly the central classes are dominant. Suppose that the class of the
element $g$ is dominant; that is, for all $h\in G$, there exists $x\in G$ such
that $g$ commutes with $h^{x}$. Then the centralizer of $g$ meets every
conjugacy class; by Jordan’s result, it is the whole of $G$, so $g\in Z(G)$ as
required. ∎

######  Problem 2.2.

Describe the dominant vertices of the NCC- or SCC-graph of a finite group.

Finally in this section, we compare our conjugacy class graphs with the
conjugacy supergraphs as defined, for example, in [9]. In these graphs, the
vertex set is the group $G$, and two vertices $g$ and $h$ are joined if and
only if there exist conjugates $g^{\prime}$ and $h^{\prime}$ of $g$ and $h$
respectively so that $\langle g^{\prime},h^{\prime}\rangle$ has the
appropriate property. These graphs are obtained from the conjugacy class
graphs defined here by “inflating” each vertex to the number of vertices in
its conjugacy class. In the other direction, we shrink a conjugacy class to a
single vertex.

It is clear that many properties of the two graphs (such as connectedness and
dominant vertices) will be unaltered by these transformations, while others
such as spectrum and clique number will change. We will not discuss this
further.

The question of when these graphs are equal is answered by the next theorem.

######  Theorem 2.6.

Let $\mathcal{P}$ be one of the properties “commutative”, “nilpotent”, or
“solvable”. A necessary and sufficient condition for the $\mathcal{P}$ graph
and the conjugacy super $\mathcal{P}$ graph to be equal is as follows.

  1. (a)

For $\mathcal{P}=\mbox{``commutative''}$: $G$ is a $2$-Engel group (one
satisfying the identity $[x,y,y]=1$).

  2. (b)

For $\mathcal{P}=\mbox{``nilpotent''}$: $G$ is nilpotent.

  3. (c)

For $\mathcal{P}=\mbox{``solvable''}$: $G$ is solvable.

###### Proof.

(a) This is [9, Theorem 2].

(b) It is clear that, if $G$ is nilpotent, then both graphs are complete. So
suppose that they are equal. Suppose that $G$ is not nilpotent. Then $G$
contains a Schmidt group (a minimal non-nilpotent group). These groups were
classified by Schmidt [69]; a convenient reference is [11].

By inspection, any such group contains a $p$-element $x$ acting non-trivially
on a $q$-group $Q$, where $p$ and $q$ are distinct primes. If $y\in Q$ with
$y^{x}\neq y$, then

$(x^{-1})^{y}x=[y,x]=y^{-1}y^{x}$

is a non-identity $q$-element. But $\langle x,x\rangle$ is nilpotent, so by
assumption $\langle x^{y},x\rangle$ is nilpotent. This is a contradiction
since all $p$-elements of a nilpotent group are contained in a single Sylow
$p$-subgroup.

(c) The key ingredient is the fact that any finite simple group can be
generated by two conjugate elements. In fact, by [39], if G is a finite simple
group, then there exists $s\in G$ such that for all nontrivial $x\in G$ there
exists $g\in G$ such that $\langle x,s^{g}\rangle=G$ (we can take $x=s$ to get
the previous claim).

It is clear that, if $G$ is solvable, then both graphs are complete. So
suppose that they are equal. Any two conjugates of an element $g$ are joined
in the $\mathcal{P}$ supergraph, and therefore are joined in the $\mathcal{P}$
graph. Suppose that $G$ is not solvable. Let $N<M<G$ be a subnormal series
such that $M/N$ is a non-abelian simple group. As noted above, there exists
$g,y\in M$ such that $M/N=\langle Ng,Ng^{y}\rangle$. In particular, $\langle
Ng,Ng^{y}\rangle$ is non-solvable and hence $\langle g,g^{y}\rangle$ is non-
solvable. However, $\langle g,g\rangle=\langle g\rangle$ is solvable, which is
a contradiction. Therefore, $G$ is solvable, completing the proof. ∎

The simplicity of the conditions in (b) and (c) compared to (a) is striking.

In the next three sections, we collate and survey some properties of the CCC-,
NCC- and SCC-graphs of finite groups.

##  3\. Commuting conjugacy class graph

We write $\mathcal{CCC}(G)$ to denote the CCC-graph of a group $G$. The CCC-
graph of $G$ is a graph whose vertex set is
$\operatorname{Cl}(G):=\\{x^{G}:x\in G\\}$, where $x^{G}$ denotes the
conjugacy class of $x$ in $G$, and two distinct vertices $a^{G}$ and $b^{G}$
are adjacent if there exist some elements $x\in a^{G}$ and $y\in b^{G}$ such
that $\langle x,y\rangle$ is an abelian group. In this section, we discuss
results on various induced subgraphs of $\mathcal{CCC}(G)$. Herzog et al. [48]
considered three induced graphs of $\mathcal{CCC}(G)$ induced by
$\operatorname{Cl}(G\setminus 1)$, $\operatorname{Cl}(G\setminus Z(G))$ and
$\operatorname{Cl}(G\setminus FC(G))$, where
$\operatorname{Cl}(S)=\\{x^{G}:x\in S\\}$ for any subset $S$ of $G$, and
$FC(G)=\\{x\in G:x^{G}\text{ is finite}\\}$ is the _FC-centre_ of $G$.

###  3.1. Connectivity of $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus 1)]$

Let $X$ be the class of groups which cannot be written as a union of
conjugates of a proper subgroup. The following theorem gives a
characterisation of residually $X$-group $G$ such that
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is complete.

######  Theorem 3.1.

[48, Proposition 1] Let $G$ be a residually $X$-group. Then the graph
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is complete if and only if
$G$ is abelian. In particular, the claim holds if $G$ is residually (finite or
solvable)-group.

Connectedness of $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is
discussed in the next two theorems.

######  Theorem 3.2.

[48, Theorem 10 and 12] Let $G$ be a finite solvable group or a periodic
solvable group. Then $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus 1)]$ has
at most two connected components, each of diameter $\leq 9$.

######  Theorem 3.3.

[48, Theorems 13-14] Let $G$ be a finite group or a locally finite group. Then
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus 1)]$ has at most six connected
components, each of diameter $\leq 19$.

The following theorems give characterisation of supersolvable/solvable groups
such that $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is disconnected.

######  Theorem 3.4.

[48, Proposition 7] Let $G$ be a supersolvable group. Then the graph
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is disconnected if and
only if $G$ is either of the groups given in the following two types:

  1. (a)

$G=A\rtimes\langle x\rangle$, where $x\in G$, $|x|=2$ and $A$ is a subgroup of
$G$ on which $x$ acts fixed-point-freely.

  2. (b)

$G$ is finite and $G=A\rtimes B$, where $A$, $B$ are non-trivial subgroups of
$G$, $A$ is nilpotent and $B$ is cyclic, and $B$ acts on $A$ fixed-point-
freely (in particular, $G$ is a Frobenius group with kernel $A$ and a cyclic
complement $B$).

######  Theorem 3.5.

[48, Theorem 16] Let $G$ be a finite solvable group such that the graph
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is disconnected. Then
there exists a nilpotent normal subgroup $H$ of $G$ such that one of the
following holds:

  1. (a)

$G=H\rtimes T$ is a Frobenius group with the kernel $H$ and a complement $T$.

  2. (b)

$G=(H\rtimes S)\rtimes\langle x\rangle$, where $S$ is a non-trivial cyclic
subgroup of $G$ of odd order which acts fixed-point-freely on $H$, $x\in
N_{G}(S)$ is such that $\langle x\rangle$ acts fixed-point-freely on $S$, and
there exist $h_{1}\in H\setminus\\{1\\}$ and $i\in\mathbb{N}$ such that
$x^{i}\neq 1$, which satisfy $[x^{i},h_{1}]=1$.

Conversely, if either (a) or (b) holds, then
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is disconnected.

###  3.2. Properties of $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$

In the following theorem Herzog et al. [48] determined all periodic groups $G$
such that $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is empty.

######  Theorem 3.6.

[48, Theorem 19] Let $G$ be a periodic non-abelian group. Then
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is empty if and only if
$G$ is isomorphic to $D_{8}$, $Q_{8}$ or $S_{3}$.

In 2016, Mohammadian et al. [56] classified all finite groups $G$ such that
the graph $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is triangle-
free and obtained the following results.

######  Theorem 3.7.

[56, Theorem 2.3] If $G$ is a finite group of odd order and the graph
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is triangle-free, then
$|G|=21$ or $27$.

######  Theorem 3.8.

[56, Theorem 3.4] Suppose $G$ is a finite group of even order which is not a
$2$-group and $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is
triangle-free. If $|Z(G)|\neq 1$, then $G$ is isomorphic to $D_{12}$ or
$T_{12}=\langle a,b:a^{4}=b^{3}=1,aba^{-1}=b^{-1}\rangle$.

######  Theorem 3.9.

[56, Theorem 3.5] If $G$ is a centreless non-solvable finite group and
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is triangle-free, then
$G$ is isomorphic to one of the groups $PSL(2,q)$ $(q\in\\{4,7,9\\})$,
$PSL(3,4)$ or $\operatorname{SmallGroup}(960,11357)$.

######  Theorem 3.10.

[56, Theorem 3.6] If $G$ is a centreless non-abelian solvable finite group and
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is triangle-free, then
$G$ is isomorphic to one of the following groups: $S_{3}$, $D_{10}$, $A_{4}$,
$S_{4}$, $\operatorname{SmallGroup}(72,41)$,
$\operatorname{SmallGroup}(192,1023)$ or
$\operatorname{SmallGroup}(192,1025)$.

######  Theorem 3.11.

[56, Theorem 3.7] If $G$ is a finite non-abelian $2$-group such that
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is triangle-free, then
$\Phi(G)\leq Z(G)$ and $C_{G}(x)=\langle x,Z(G)\rangle$ whenever $x\in
G\setminus Z(G)$. Furthermore, either $G\cong D_{8}$, $G\cong Q_{8}$ or
$|G:Z(G)|=|Z(G)|$.

###  3.3. Structure of $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$

In [67, 68, 66] structures of commuting conjugacy class graphs of certain
finite non-abelian groups were determined. In this section, we shall discuss
the structures of CCC-graphs of dihedral group, generalized quaternion group,
semi-dihedral group, the groups $U_{(n,m)}$, $V_{8n}$ and $G(p,m,n)$ along
with some other groups such that
$\frac{G}{Z(G)}\cong\mathbb{Z}_{p}\times\mathbb{Z}_{p}$ or $D_{2n}$, where $p$
is a prime and $D_{2n}=\langle
x,y:~{}x^{n}=y^{2}=1,~{}yxy^{-1}=x^{-1}\rangle$.

######  Theorem 3.12.

[66, Theorem 1.2] Let $G$ be a finite group with centre $Z(G)$ and
$\frac{G}{Z(G)}$ is isomorphic to the dihedral group $D_{2n}$. Then

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))]=\begin{cases}K_{\frac{(n-1)|Z(G)|}{2}}\cup
2K_{\frac{|Z(G)|}{2}},&\text{for }2\mid n\\\ K_{\frac{(n-1)|Z(G)|}{2}}\cup
K_{|Z(G)|},&\text{for }2\nmid n.\end{cases}$

As a corollary to Theorem 3.12, we get the structure of
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ when $G$ is the
dihedral group $D_{2n}$, the generalized quaternion group $Q_{4m}=\langle
x,y:~{}x^{2m}=1,x^{m}=y^{2},y^{-1}xy=x^{-1}\rangle$, the semi-dihedral group
$SD_{8n}=\langle x,y:~{}x^{4n}=y^{2}=1,yxy=x^{2n-1}\rangle$ the group
$U_{(n,m)}=\langle x,y:~{}x^{2n}=y^{m}=1,~{}x^{-1}yx=y^{-1}\rangle$ and the
group $U_{6n}=\langle x,y:~{}x^{2n}=y^{3}=1,~{}x^{-1}yx=y^{-1}\rangle$.
Further, Salahshour and Ashrafi [68, Proposition 2.4 and 2.6] determined the
structures of $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ when $G$
is the group $V_{8n}=\langle x,y:$
$~{}x^{2n}=y^{4}=1,~{}yx=x^{-1}y^{-1},~{}y^{-1}x=x^{-1}y\rangle$ and the group
$G(p,m,n)=$ $\langle
x,y:~{}x^{p^{m}}=y^{p^{n}}=[x,y]^{p}=1,[x,[x,y]]=[y,[x,y]]=1\rangle$ as given
below:

$\mathcal{CCC}(V_{8n})[\operatorname{Cl}(V_{8n}\setminus
Z(V_{8n}))]=\begin{cases}K_{2n-2}\cup 2K_{2},&\text{for }2\mid n\\\
K_{2n-1}\cup 2K_{1},&\text{for }2\nmid n\end{cases}$

and

$\displaystyle\mathcal{CCC}(G(p,m,n))[\operatorname{Cl}(G(p,m,n)$
$\displaystyle\setminus Z(G(p,m,n)))]$
$\displaystyle=2K_{p^{m+n-1}-p^{m+n-2}}\cup(p^{n}-p^{n-1})K_{p^{m-n}(p^{n}-p^{n-1})}.$

Note that all the groups considered above are AC-groups (non-abelian groups
whose centralizers of non-central elements are abelian). Salahshour and
Ashrafi [68] also obtained the structure of
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ if $G$ is a finite AC-
group. Let $\operatorname{Cent}(G)=\\{C_{G}(a):a\in G\\}$, where $C_{G}(a)$ is
the centralizer of $a\in G$. Consider the equivalence relation $\sim$ on
$\operatorname{Cent}(G)\setminus\\{G\\}$ given by $C_{G}(a)\sim C_{G}(b)$ if
and only if $C_{G}(a)$ and $C_{G}(b)$ are conjugate in $G$. Then we have the
following result.

######  Theorem 3.13.

[68, Theorem 3.3] Let $G$ be a finite AC-group with centre $Z(G)$. Then

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))]=\bigcup\limits_{\frac{C_{G}(a)}{\sim}\in
EC(G)}K_{n_{\frac{C_{G}(a)}{\sim}}}$

where $EC(G)=\frac{Cent(G)\setminus\\{G\\}}{\sim}$ is the set of all
equivalence classes of $\sim$ and
$n_{\frac{C_{G}(a)}{\sim}}=\frac{|C_{G}(a)|-|Z(G)|}{[N_{G}(C_{G}(a)):C_{G}(a)]}$.

Salahshour and Ashrafi [67] determined the structures of
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ when $G$ is a finite
non-abelian group such that $\frac{G}{Z(G)}$ has order $p^{2}$ or $p^{3}$ as
given in the following theorems.

######  Theorem 3.14.

[67, Theorem 3.1] Let $G$ be a finite non-abelian group with centre $Z(G)$ and
$\frac{G}{Z(G)}\cong\mathbb{Z}_{p}\times\mathbb{Z}_{p}$, where $p$ is prime.
Then $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]=(p+1)K_{n}$, where
$n=\frac{(p-1)|Z(G)|}{p}$.

######  Theorem 3.15.

[67, Theorem 3.3] Let $G$ be a finite non-abelian group with centre $Z(G)$ and
$|\frac{G}{Z(G)}|=p^{3}$, where $p$ is a prime. Then one of the following is
satisfied:

  1. (a)

If $\frac{G}{Z(G)}$ is abelian then
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]=K_{m}\cup p^{2}K_{n}$ or
$(p^{2}+p+1)K_{n}$, where $m=\frac{(p^{2}-1)|Z(G)|}{p}$ and
$n=\frac{(p-1)|Z(G)|}{p^{2}}$.

  2. (b)

If $\frac{G}{Z(G)}$ is non-abelian then
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]=K_{m}\cup
kpK_{n_{1}}\cup(p-k)K_{n_{2}}$, $(kp+1)K_{n_{1}}\cup(p+1-k)K_{n_{2}}$,
$K_{m}\cup pK_{n_{2}}$, $(p^{2}+p+1)K_{n_{1}}$ or
$K_{n_{1}}\cup(p+1)K_{n_{2}}$, where $m=\frac{(p^{2}-1)|Z(G)|}{p}$,
$n_{1}=\frac{(p-1)|Z(G)|}{p^{2}}$, $n_{2}=\frac{(p-1)|Z(G)|}{p}$, $1\leq k\leq
p$.

As a corollary, it follows that $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))]=(p+1)K_{p(p-1)}$ or $K_{(p^{2}-1)}\cup pK_{p-1}$ if $G$ is a non-
abelian $p$-group of order $p^{4}$. Ashrafi and Salahshour [10, Theorem 1.2]
also obtained the structure of $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))]$ when $\frac{G}{Z(G)}$ is isomorphic to
$\mathbb{Z}_{p^{2}}\rtimes\mathbb{Z}_{p^{2}}$, where $p$ is a prime. In a
recent work, Rezaei and Foruzanfar [65] have determined the structure of
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ when $\frac{G}{Z(G)}$
is isomorphic to a Frobenius group of order $pq$ or $p^{2}q$, where $p,q$ are
primes.

###  3.4. Genus of $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$

For any graph $\Gamma$, we write $\gamma(\Gamma)$ to denote its genus. The
genus of $\Gamma$ is the smallest integer $k\geq 0$ such that $\Gamma$ can be
embedded on the surface obtained by attaching $k$ handles to a sphere. If
$\gamma(\Gamma)$ is equal to $0,1,2$, or $3$, then $\Gamma$ is called planar,
toroidal, double-toroidal, or triple-toroidal, respectively. Clearly,
$\gamma(K_{1})=\gamma(K_{2})=0$. For $n\geq 3$, by [75, Theorem 6-38], we have

$\gamma(K_{n})=\bigg{\lceil}\frac{(n-3)(n-4)}{12}\bigg{\rceil},$

where $\lceil a\rceil$ denotes the smallest integer greater than or equal to
$a$ for any real number $a$. It is worth mentioning that finite non-abelian
groups $G$ for which $\mathcal{C}(G)[G\setminus Z(G)]$ (the induced subgraph
of commuting graph of $G$ induced by $G\setminus Z(G)$) is planar have been
characterised (see [6, Theorem 2.2]), toroidal (see [6, Theorem 2.2] and [28,
Theorem 3.3]), double-toroidal (see [63, Theorem 3.3]) and triple-toroidal
(see [63, Theorem 3.7]). In this regard, we have the following problem.

######  Problem 3.1.

Characterize all finite non-abelian groups $G$ such that the induced subgraph
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ of $\mathcal{CCC}(G)$
is planar, toroidal, double-toroidal or triple-toroidal.

This problem was considered by Bhowal and Nath [15] and they characterised the
dihedral groups, generalized quaternion groups and semidihedral groups such
that $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is planar,
toroidal, double-toroidal or triple-toroidal. We have the following theorems
for instance.

######  Theorem 3.16.

[16, Theorem 2.2] Let $G$ be the dihedral group $D_{2n}$. Then

  1. (a)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is planar if and only
if $3\leq n\leq 10$.

  2. (b)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is toroidal if and only
if $11\leq n\leq 16$.

  3. (c)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is double-toroidal if
and only if $n=17,18$.

  4. (d)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is triple-toroidal if
and only if $n=19,20$.

  5. (e)

$\gamma(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\begin{cases}\left\lceil\frac{(n-7)(n-9)}{48}\right\rceil,&\text{ for
$2\nmid n$ and $n\geq 21$ }\vspace{.2cm}\\\
\left\lceil\frac{(n-8)(n-10)}{48}\right\rceil,&\text{ for $2\mid n$ and $n\geq
22$. }\end{cases}$

######  Theorem 3.17.

[16, Theorem 2.4] Let $G$ be the generalized quaternion group $Q_{4m}$. Then

  1. (a)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is planar if and only
if $m=2,3,4$ or $5$.

  2. (b)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is toroidal if and only
if $m=6,7$ or $8$.

  3. (c)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is double-toroidal if
and only if $m=9$.

  4. (d)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is triple-toroidal if
and only if $m=10$.

  5. (e)

$\gamma(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\left\lceil\frac{(m-4)(m-5)}{12}\right\rceil$ for $m\geq 11$.

######  Theorem 3.18.

[16, Theorem 2.3] Let $G$ be the semidihedral group $SD_{8n}$. Then

  1. (a)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is planar if and only
if $n=2$ or $3$.

  2. (b)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is toroidal if and only
if $n=4$.

  3. (c)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is double-toroidal if
and only if $n=5$.

  4. (d)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is not triple-toroidal.

  5. (e)

$\gamma(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\begin{cases}\left\lceil\frac{(n-3)(2n-5)}{6}\right\rceil,&\text{ for
$2\nmid n$ and $n\geq 7$ }\vspace{.2cm}\\\
\left\lceil\frac{(n-2)(2n-5)}{6}\right\rceil,&\text{ for $2\mid n$ and $n\geq
6$. }\end{cases}$

Bhowal and Nath [16] also considered the groups $V_{8n}$, $U_{(n,m)}$ and
$G(p,m,n)$ in their study and obtained the following result.

######  Theorem 3.19.

[16, Corollary 2.8] Let $G$ be a group isomorphic to $D_{2n}$, $SD_{8n}$,
$Q_{4m}$, $V_{8n}$, $U_{(n,m)}$ or $G(p,m,n)$. Then

  1. (a)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is planar if and only
if $G=D_{6},D_{8},D_{10},D_{12},D_{14}$, $D_{16}$, $D_{18},D_{20},$ $SD_{16}$,
$SD_{24}$, $Q_{8},Q_{12},Q_{16},Q_{20},V_{16},U_{(2,2)}$, $U_{(2,3)}$,
$U_{(2,4)}$, $U_{(2,5)}$, $U_{(2,6)}$, $U_{(3,2)}$, $U_{(3,3)}$, $U_{(3,4)}$,
$U_{(4,2)}$, $U_{(4,3)}$, $U_{(4,4)}$, $G(2,1,1)$, $G(3,1,1)$, $G(5,1,1)$,
$G(2,2,1)$, $G(2,3,1)$, $G(2,1,2)$, $G(2,2,2)$ or $G(2,1,3)$.

  2. (b)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is toroidal if and only
if $G=D_{22}$, $D_{24}$, $D_{26}$, $D_{28}$, $D_{30}$, $D_{32}$, $SD_{32}$,
$Q_{24}$, $Q_{28}$, $Q_{32}$, $V_{24}$, $V_{32}$, $U_{(2,7)}$, $U_{(2,8)}$,
$U_{(3,5)}$ or $U_{(3,6)}$.

  3. (c)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is double-toroidal if
and only if $G=D_{34}$, $D_{36}$, $SD_{40}$, $Q_{36}$, $U_{(2,9)}$,
$U_{(2,10)}$, $U_{(4,5)}$, $U_{(4,6)}$, $U_{(5,2)}$, $U_{(5,3)}$, $U_{(6,2)}$,
$U_{(6,3)}$, $U_{(7,2)}$, $U_{(7,3)}$ or $G(3,1,2)$.

  4. (d)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is triple-toroidal if
and only if $G=D_{38}$, $D_{40}$, $Q_{40}$, $V_{40}$, $U_{(3,7)}$,
$U_{(3,8)}$, $U_{(5,4)}$, $U_{(6,4)}$ or $U_{(7,4)}$.

It may be interesting to continue similar study for the groups with
known/unknown structures of $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))]$ and answer Problem 3.1.

###  3.5. Various spectra and energies of
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$

The spectrum of a finite graph $\Gamma$ with vertex set $V(\Gamma)$, denoted
by $\operatorname{Spec}(\Gamma)$, is the set of eigenvalues of its adjacency
matrix with multiplicities. If
$\operatorname{Spec}(\Gamma)=\\{\alpha_{1}^{a_{1}},\alpha_{2}^{a_{2}},\ldots,\alpha_{k}^{a_{k}}\\}$
for some $\Gamma$ then we mean that $\alpha_{1},\alpha_{2},\ldots,\alpha_{k}$
are the eigenvalues of the adjacency matrix of $\Gamma$ with multiplicities
$a_{1},a_{2},\ldots,a_{k}$ respectively. Similarly,
$\operatorname{L-spec}-spec(\Gamma)$ and $\operatorname{Q-spec}-spec(\Gamma)$
denote the Laplacian spectrum (L-spectrum) and signless Laplacian spectrum
(Q-spectrum) of $\Gamma$ i.e., the set of eigenvalues of the Laplacian and
signless Laplacian matrices of $\Gamma$ respectively. A graph $\Gamma$ is
called integral/L-integral/Q-integral if
$\operatorname{Spec}(\Gamma)$/$\operatorname{L-spec}-spec(\Gamma)$/$\operatorname{Q-spec}-spec(\Gamma)$
contains only integers. To determine all the integral/ L-integral/Q-integral
graphs is a general problem in graph theory. Various spectra of
$\mathcal{C}(G)[G\setminus Z(G)]$ were computed in [28, 29, 30, 59] and
obtained various groups such that $\mathcal{C}(G)[G\setminus Z(G)]$ is
integral/L-integral/Q-integral. Note that the following problem is still open.

######  Problem 3.2.

Determine all the finite non-abelian groups $G$ such that
$\mathcal{C}(G)[G\setminus Z(G)]$ is integral/L-integral/Q-integral.

The energy, Laplacian energy (L-energy) and signless Laplacian energy
(Q-energy) of $\Gamma$ denoted by $E(\Gamma)$, $LE(\Gamma)$ and
$LE^{+}(\Gamma)$ respectively are given by

$E(\Gamma):=\sum\limits_{\alpha\in\operatorname{Spec}(\Gamma)}|\alpha|,\qquad
LE(\Gamma):=\sum\limits_{\beta\in\operatorname{L-spec}-spec(\Gamma)}\bigg{|}\beta-\frac{tr(\mathcal{D}(\Gamma))}{|V(\Gamma)|}\bigg{|}$
$\text{ and
}LE^{+}(\Gamma):=\sum\limits_{\gamma\in\operatorname{Q-spec}-spec(\Gamma)}\bigg{|}\gamma-\frac{tr(\mathcal{D}(\Gamma))}{|V(\Gamma)|}\bigg{|},$

where $\mathcal{D}(\Gamma)$ is the degree matrix of $\Gamma$ and
$tr(\mathcal{D}(\Gamma))$ is the trace of $\mathcal{D}(\Gamma)$. In 1978,
Gutman [41] introduced the notion of energy of a graph. In Huckel theory,
$\pi$-electron energy of a conjugated carbon molecule is approximated by
$E(\mathcal{G})$. Subsequently, Gutman and Zhou [46] in 2006 and Abreua et al.
[5] in 2008 introduced L-energy and Q-energy of a graph. Applications of these
energies can be found in crystallography, theory of macromolecules, analysis
and comparison of protein sequences, network analysis, satellite
communication, image analysis and processing etc. (see [45] and the references
therein). In 2009, Gutman et al. [44] conjectured (E-LE conjecture) that

(3) $E(\Gamma)\leq LE(\Gamma).$

Though (3) was disproved in [55, 71] people wanted to know whether this
conjecture is true for various graphs defined on groups. The following problem
for $\mathcal{C}(G)[G\setminus Z(G)]$ is considered in [26, 31].

######  Problem 3.3.

Determine all the finite non-abelian groups $G$ such that
$\mathcal{C}(G)[G\setminus Z(G)]$ satisfy the following inequalities:

  1. (a)

$E(\mathcal{C}(G)[G\setminus Z(G)])\leq LE(\mathcal{C}(G)[G\setminus Z(G)])$.

  2. (b)

$LE^{+}(\mathcal{C}(G)[G\setminus Z(G)])\leq LE(\mathcal{C}(G)[G\setminus
Z(G)])$.

A graph $\Gamma$ is called hyperenergetic, L-hyperenergetic and
Q-hyperenergetic if $E(\Gamma)$ $>E(K_{|V(\Gamma)|})$,
$LE(\Gamma)>LE(K_{|V(\Gamma)|})$ and $LE^{+}(\Gamma)>LE^{+}(K_{|V(\Gamma)|})$
respectively. The concept of hyperenergetic graph was given by Walikar et al.
[74] and Gutman [42], independently in 1999. The concept of L-hyperenergetic
and Q-hyperenergetic graph can be found in [33]. Again, $\Gamma$ is called
borderenergetic (introduced by Gong et al. [36]), L-borderenergetic
(introduced by Tura [73]) and Q-borderenergetic (introduced by Tao et al.
[72]) if $E(\Gamma)=E(K_{|V(\Gamma)|})$, $LE(\Gamma)=LE(K_{|V(\Gamma)|})$ and
$LE^{+}(\Gamma)=LE^{+}(K_{|V(\mathcal{G})|})$ respectively. The following
conjecture was posed by Gutman [41] in 1978.

######  Conjecture 3.20.

Any finite graph $\Gamma\ncong K_{|v(\mathcal{G})|}$ is non-hyperenergetic.

This conjecture was also disproved by different mathematicians providing
counter examples (see [43]). However, the search for counter examples to
Conjecture 3.20 continued. In [70], the following problem for
$\mathcal{C}(G)[G\setminus Z(G)]$ was considered.

######  Problem 3.4.

Determine all the finite non-abelian groups $G$ such that
$\mathcal{C}(G)[G\setminus Z(G)]$
is hyperenergetic/ borderenergetic/ L-hyperenergetic/ L-borderenergetic/
Q-hyper-
energetic/ Q-borderenergetic.

In [15, 17], Bhowal and Nath considered problems corresponding to the Problems
3.2–3.4 for $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$. They
considered the dihedral groups, generalized quaternion groups and semidihedral
groups and obtained the following results.

######  Theorem 3.21.

[15, Theorem 3.1] If $G$ is the dihedral group $D_{2n}$, then

  1. (a)

$\operatorname{Spec}(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))])$

$=\begin{cases}\left\\{(0)^{1},(-1)^{\frac{n-3}{2}},\left(\frac{n-3}{2}\right)^{1}\right\\},&\text{for
}2\nmid n\\\
\left\\{(0)^{2},(-1)^{\frac{n}{2}-2},\left(\frac{n}{2}-2\right)^{1}\right\\},&\text{
for }2\mid n\text{ and }2\mid\frac{n}{2}\\\
\left\\{(-1)^{\frac{n}{2}-1},(1)^{1},\left(\frac{n}{2}-2\right)^{1}\right\\},&\text{
for }2\mid n\text{ and }2\nmid\frac{n}{2}\end{cases}$

and $E(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\begin{cases}n-3,&\text{ for }2\nmid n\\\ n-4,&\text{ for }2\mid
n\text{ and }2\mid\frac{n}{2}\\\ n-2,&\text{ for }2\mid n\text{ and
}2\nmid\frac{n}{2}.\end{cases}$

  2. (b)

$\operatorname{L-spec}-spec(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])$

$=\begin{cases}\left\\{(0)^{2},\left(\frac{n-1}{2}\right)^{\frac{n-3}{2}}\right\\},&\text{
for }2\nmid n\\\
\left\\{(0)^{3},\left(\frac{n}{2}-1\right)^{\frac{n}{2}-2}\right\\},&\text{
for }2\mid n\text{ and }2\mid\frac{n}{2}\\\
\left\\{(0)^{2},2^{1},\left(\frac{n}{2}-1\right)^{\frac{n}{2}-2}\right\\},&\text{
for }2\mid n\text{ and }2\nmid\frac{n}{2}\end{cases}$

and $LE(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])\\!=\\!\begin{cases}\frac{2(n-1)(n-3)}{n+1},&\text{ for }2\nmid n\\\
\frac{3(n-2)(n-4)}{n+2},&\text{ for }2\mid n\text{ and }2\mid\frac{n}{2}\\\
4,&\text{ for $n=6$}\\\ \frac{(n-4)(3n-10)}{n+2},&\text{ for }2\mid n,~{}n\geq
10\\\ &\qquad\text{ and }2\nmid\frac{n}{2}.\end{cases}$

  3. (c)

$\operatorname{Q-spec}-spec(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])$

$=\begin{cases}\left\\{(0)^{1},(n-3)^{1},\left(\frac{n-5}{2}\right)^{\frac{n-3}{2}}\right\\},&\text{
for }2\nmid n\\\
\left\\{(0)^{2},(n-4)^{1},\left(\frac{n}{2}-3\right)^{\frac{n}{2}-2}\right\\},&\text{
for }2\mid n\text{ and }2\mid\frac{n}{2}\\\
\left\\{(0)^{1},(2)^{1},(n-4)^{1},\left(\frac{n}{2}-3\right)^{\frac{n}{2}-2}\right\\},&\text{
for }2\mid n\text{ and }2\nmid\frac{n}{2}\end{cases}$

 and $LE^{+}(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\begin{cases}\frac{(n-3)(n+3)}{n+1},&\text{ for }2\nmid n\\\
\frac{(n-4)(n+6)}{n+2},&\text{ for }n=4,8\\\ \frac{2(n-2)(n-4)}{n+2},&\text{
for }2\mid n,\frac{n}{2}\text{ and }n\geq 12\\\ 4,&\text{ for $n=6$}\\\
\frac{22}{3},&\text{ for $n=10$}\\\ \frac{2(n-2)(n-6)}{n+2},&\text{ for }2\mid
n,~{}2\nmid\frac{n}{2}\\\ &\qquad\text{ and }n\geq 14.\end{cases}$

######  Theorem 3.22.

[15, Theorem 3.2] If $G$ is the generalized quaternion group $Q_{4m}$, then

  1. (a)

$\operatorname{Spec}(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\begin{cases}\left\\{(-1)^{m-1},(1)^{1},(m-2)^{1}\right\\},&\text{ for
}2\nmid m\\\ \left\\{(-1)^{m-2},(0)^{2},(m-2)^{1}\right\\},&\text{ for }2\mid
m\end{cases}$

and $E(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\begin{cases}2m-2,&\text{ for }2\nmid m\\\ 2m-4,&\text{ for }2\mid
m.\end{cases}$

  2. (b)

$\operatorname{L-spec}-spec(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\begin{cases}\left\\{(0)^{2},(2)^{1},(m-1)^{m-2}\right\\},&\text{ for
}2\nmid m\\\ \left\\{(0)^{3},(m-1)^{m-2}\right\\},&\text{ for }2\mid
m\end{cases}$

 and $LE(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\begin{cases}4,&\text{ for $m=3$}\\\ \frac{2(m-2)(3m-5)}{m+1},&\text{
for }2\nmid m\text{ and }m\geq 5\\\ \frac{6(m-1)(m-2)}{m+1},&\text{ for }2\mid
m.\end{cases}$

  3. (c)

$\operatorname{Q-spec}-spec(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])$

$\qquad\qquad=\begin{cases}\left\\{(2)^{1},(0)^{1},(2m-4)^{1},(m-3)^{m-2}\right\\},&\text{for
}2\nmid m\\\ \left\\{(0)^{2},(2m-4)^{1},(m-3)^{m-2}\right\\},&\text{for }2\mid
m\end{cases}$

 and $LE^{+}(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\begin{cases}4,&\text{ for $m=3$}\\\ \frac{22}{3},&\text{ for
$m=5$}\\\ \frac{4(m-1)(m-3)}{m+1},&\text{ for }2\nmid m\text{ and }m\geq 7\\\
\frac{2(m-2)(m+3)}{m+1},&\text{ for $m=2,4$}\\\
\frac{4(m-1)(m-2)}{m+1},&\text{ for }2\mid m\text{ and }m\geq 6.\end{cases}$

######  Theorem 3.23.

[15, Theorem 3.5] If $G$ is the semidihedral group $SD_{8n}$, then

  1. (a)

$\operatorname{Spec}(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\begin{cases}\left\\{(-1)^{2n},(3)^{1},\left(2n-3\right)^{1}\right\\},&\text{
for $2\nmid n$}\\\
\left\\{(-1)^{2n-2},(0)^{2},\left(2n-2\right)^{1}\right\\},&\text{ for $2\mid
n$}\end{cases}$

and $E(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\begin{cases}4n,&\text{ for $2\nmid n$}\\\ 4n-4,&\text{ for $2\mid
n$}.\end{cases}$

  2. (b)

$\operatorname{L-spec}-spec(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\begin{cases}\left\\{(0)^{2},(4)^{3},\left(2n-2\right)^{2n-3}\right\\},&\text{
for $2\nmid n$}\\\ \left\\{(0)^{3},\left(2n-1\right)^{2n-2}\right\\},&\text{
for $2\mid n$}\end{cases}$

and $LE(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\begin{cases}12,&\text{ for $n=3$}\\\
\frac{2(2n-3)(5n-11)}{n+1},&\text{ for $2\nmid n$ and $n\geq 5$}\\\
\frac{6(2n-1)(2n-2)}{2n+1},&\text{ for $2\mid n$.}\end{cases}$

  3. (c)

$\operatorname{Q-spec}-spec(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])$

$\qquad\qquad=\begin{cases}\left\\{(6)^{1},(2)^{3},(4n-6)^{1},\left(2n-4\right)^{2n-3}\right\\},&\text{
for $2\nmid n$}\\\
\left\\{(0)^{2},(4n-4)^{1},\left(2n-3\right)^{2n-2}\right\\},&\text{ for
$2\mid n$}\end{cases}$

and $LE^{+}(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))])=\begin{cases}12,&\text{ for $n=3$}\\\ 22,&\text{ for $n=5$}\\\
\frac{16(n-1)(n-3)}{n+1},&\text{ for $2\nmid n$ and $n\geq 7$}\\\
\frac{28}{5},&\text{ for $n=2$}\\\ \frac{4(2n-1)(2n-2)}{2n+1},&\text{ for
$2\mid n$ and $n\geq 4$.}\end{cases}$

Bhowal and Nath [15, 17] also considered the groups $V_{8n}$, $U_{(n,m)}$ and
$G(p,m,n)$ in their study and obtained the following results.

######  Theorem 3.24.

[15, Corollary 3.6] If $G$ is isomorphic to $D_{2n}$, $Q_{4m}$, $SD_{8n}$,
$V_{8n}$, $U_{(n,m)}$ and $G(p,m,n)$ then
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is integral, L-integral
and Q-integral.

Comparing various energies of $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
Z(G))]$ they also obtained the following results.

######  Theorem 3.25.

[15, Theorem 4.6] Let $G$ be a finite non-abelian group and
$\Gamma=\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$. Then

  1. (a)

$E(\Gamma)=LE^{+}(\Gamma)=LE(\Gamma)$ if $G$ is isomorphic to
$D_{6},D_{8},D_{12},Q_{8},Q_{12}$, $U_{(n,2)}$, $U_{(n,3)},U_{(n,4)}$ $(n\geq
2)$, $V_{16}$, $SD_{24}$ or $G(p,m,1)$ $(p\geq 2,m\geq 1)$.

  2. (b)

$LE^{+}(\Gamma)<E(\Gamma)<LE(\Gamma)$ if $G$ is isomorphic to $D_{20},Q_{20}$,
$U_{(2,5)},U_{(3,5)}$, $U_{(2,6)}$ or $G(2,2,2)$.

  3. (c)

$E(\Gamma)<LE^{+}(\Gamma)<LE(\Gamma)$ if $G$ is isomorphic to
$D_{14},D_{16},D_{18}$, $D_{2n}$ $(n\geq 11)$, $Q_{16},Q_{24},Q_{4m}$ $(m\geq
8)$, $U_{(n,5)}$ $(n\geq 4)$, $U_{(n,m)}$ $(m\geq 6\text{ and }n\geq 3)$,
$U_{(n,m)}$ $(m\geq 8\text{ and }n\geq 2)$, $V_{8n}$ $(n\geq 3)$, $SD_{16}$,
$SD_{8n}$ $(n\geq 4)$, $G(2,m,2)$ $(m\geq 3)$, $G(p,m,2)$ $(p\geq 3,m\geq 1)$
or $G(p,m,n)$ $(n\geq 3,p\geq 2,m\geq 1)$.

  4. (d)

$E(\Gamma)=LE^{+}(\Gamma)<LE(\Gamma)$ if $G$ is isomorphic to $Q_{28}$ or
$U_{(2,7)}$

  5. (e)

$E(\Gamma)<LE^{+}(\Gamma)$ $=LE(\Gamma)$ if $G$ is isomorphic to $D_{10}$ and
$G(2,1,2)$.

######  Theorem 3.26.

[15, Theorem 5.6] Let $G$ be a finite non-abelian group. Then

  1. (a)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is neither
hyperenergetic, borderenergetic,
L-hyperenergetic, L- borderenergetic, Q-hyperenergetic nor Q-borderenergetic
if $G$ is isomorphic to $D_{8}$, $D_{12}$, $D_{2n}\,(\text{$n$ is odd})$,
$Q_{8}$, $Q_{12}$, $Q_{16}$, $U_{(2,6)}$, $U_{(n,3)}$, $U_{(n,4)}$ $(n\geq
2)$, $V_{16}$, $SD_{16}$, $SD_{24}$, $G(p,m,1)\,\,(p\geq 2$ and $m\geq 1)$,
$G(2,1,2)$ or $G(2,2,2)$.

  2. (b)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is L-borderenergetic
but neither hyperenergetic, borderenergetic, L-borderenergetic,
Q-hyperenergetic nor Q-borderenergetic if $G$ is isomorphic to $Q_{20}$ or
$U_{(2,5)}$.

  3. (c)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is L-hyperenergetic but
neither hyperenergetic, borderenergetic, L- borderenergetic, Q-hyperenergetic
nor Q-borderenergetic if $G$ is isomorphic to $D_{16}$, $D_{20}$, $D_{24}$,
$D_{28}$, $Q_{24}$, $Q_{28}$, $U_{(3,5)}$, $U_{(3,6)}$, $U_{(2,7)}$, $V_{24}$,
$V_{32}$, $G(2,3,2)$ or $G(2,1,3)$.

  4. (d)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is L-hyperenergetic and
Q-borderenergetic but neither hyperenergetic, borderenergetic,
L-borderenergetic nor Q-hyperenergetic if $G$ is isomorphic to $SD_{40}$.

  5. (e)

$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is L-hyperenergetic and
Q-hyperenergetic but neither hyperenergetic, borderenergetic,
L-borderenergetic nor Q-borderenergetic if $G$ is isomorphic to $D_{2n}$ ($n$
is even, $n\geq 16$), $Q_{4m}\,\,(m\geq 8)$, $U_{(n,5)}\,\,(n\geq 4)$,
$U_{(n,6)}\,\,(n\geq 4)$, $U_{(n,7)}\,\,(n\geq 3)$, $U_{(n,m)}$ $(n\geq 2$ and
$m\geq 8)$, $V_{8n}$ $(n\geq 5)$, $SD_{32}$, $SD_{8n}\,\,(n\geq 6)$,
$G(2,m,2)\,\,(m\geq 4)$, $G(p,m,2)$ $(p\geq 3$ and $m\geq 1)$,
$G(2,m,3)\,\,(m\geq 2)$ or $G(p,m,n)\,\,(n\geq 4,p\geq 2\text{ and }m\geq 1)$.

######  Theorem 3.27.

[15, Theorem 5.7] Let $G$ be a finite non-abelian group.

  1. (a)

Then $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is neither
hyperenergetic nor borderenergetic, for $G$ is isomorphic to $D_{2n}$,
$Q_{4m}$, $U_{(n,m)}$, $V_{8n}$, $SD_{8n}$ or $G(p,m,n)$.

  2. (b)

Then $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is
L-hyperenergetic, for $G$ is isomorphic to $D_{2n}$ $(n$ is even, $n\geq 8)$,
$Q_{4m}$ $(m\geq 6)$, $U_{(n,5)}$ $(n\geq 3)$, $U_{(n,6)}$ $(n\geq 3)$,
$U_{(n,m)}$ $(n\geq 2\text{ and }m\geq 7)$, $V_{8n}$ $(n\geq 3)$, $SD_{8n}$
$(n\geq 4)$, $G(2,m,2)$ $(m\geq 3)$, $G(p,m,2)$ $(p\geq 3\text{ and }m\geq
1)$, $G(2,m,3)$ $(m\geq 1)$ or $G(p,m,n)$ $(n\geq 4,p\geq 2$ and $m\geq 1)$.

  3. (c)

Then $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is
L-borderenergetic, for $G$ is isomorphic to $Q_{20}$ or $U_{(2,5)}$.

  4. (d)

Then $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is
Q-hyperenergetic, for $G$ is isomorphic to $D_{2n}$ $n$ is even, $n\geq 16)$,
$Q_{4m}$ $(m\geq 8)$, $U_{(n,5)}$ $(n\geq 4)$, $U_{(n,6)}$ $(n\geq 4)$,
$U_{(n,7)}$ $(n\geq 3)$, $U_{(n,m)}$ $(n\geq 2\text{ and }m\geq 8)$,
$V_{8n}\,\,(n\geq 5)$, $SD_{32}$, $SD_{8n}$ $(n\geq 6)$, $G(2,m,2)$ $(m\geq
4)$, $G(p,m,2)$ $(p\geq 3\text{ and }m\geq 1)$, $G(2,m,3)$ $(m\geq 2)$ or
$G(p,m,n)$ $(n\geq 4$, $p\geq 2$ and $m\geq 1)$.

  5. (e)

Then $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ is
Q-borderenergetic, for $G$ is isomorphic to $SD_{40}$.

We conclude this section noting that problems analogous to Problems 3.2 – 3.4
for various common neighborhood spectrum and energies of
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ were considered in [50,
51]. Common neighborhood spectrum/energy, common neighborhood Laplacian
spectrum/energy and common neighborhood signless Laplacian spectrum/energy of
graphs were introduced in [8] and [52]. Various common neighborhood spectrum
and energies of $\mathcal{C}(G)[G\setminus Z(G)]$ were considered in [33, 34,
61].

###  3.6. Properties of $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
FC(G))]$

The FC-centre of a group $G$ is the set of elements $x\in G$ such that $x^{G}$
is finite. Herzog et al. [48] obtained the following results for the induced
subgraph $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus FC(G))]$ of
$\mathcal{CCC}(G)$ induced by $\operatorname{Cl}(G)\setminus\\{g^{G}:g\in
FC(G)\\}$ when $G$ is a periodic group.

######  Theorem 3.28.

[48, Theorem 22] Let $G$ be a periodic group such that the graph
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus FC(G))]$ is empty. Write
$F=FC(G)$ and suppose that there exists $xF\in G/F$ such that $|xF|=3$. Then
$G$ has the following structure:

$G=F\rtimes\langle a,b\rangle,$

where $|a|=3$, $|b|=2$, $a^{b}=a^{-1}$ $(\text{i.e.}\langle a,b\rangle\cong
S_{3})$, $F$ is an elementary abelian $2$-group, $F=D_{1}\times D_{2}$, where
$D_{1}=\\{[d,b]:d\in F\\}$, $D_{2}=\\{[d,ab]:d\in F\\}$, $D_{1}$ and $D_{2}$
are infinite subgroups of $F$ and $a$ acts fixed-point-freely on $F$.

Conversely, if $G$ has the above structure, then $F=FC(G)$ and the graph
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus FC(G))]$ is empty.

######  Theorem 3.29.

[48, Theorem 23] Let $G$ be a periodic group such that the graph
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus FC(G))]$ is empty. Write
$F=FC(G)$. Then $G$ is locally finite and either $G$ is a hypercentral
$2$-group with $G/F$ abelian of exponent $2$ or $G/F$ is finite. In the latter
case, either $G/F$ is a finite elementary abelian $2$-group or $G/F\cong
S_{3}$ and $G$ has the structure described in Theorem 3.28.

##  4\. Nilpotent conjugacy class graph

We write $\mathcal{NCC}(G)$ to denote the NCC-graph of a group $G$. The NCC-
graph of $G$ is a graph whose vertex set is $\operatorname{Cl}(G)$ and two
distinct vertices $a^{G}$ and $b^{G}$ are adjacent if there exist some
elements $x\in a^{G}$ and $y\in b^{G}$ such that $\langle x,y\rangle$ is a
nilpotent group. In this section, we discuss various results on induced
subgraphs of $\mathcal{NCC}(G)$. Mohammadian and Erfanian [57] considered two
induced subgraphs of $\mathcal{NCC}(G)$ induced by
$\operatorname{Cl}(G\setminus 1)$ and
$\operatorname{Cl}(G\setminus\operatorname{Nil}(G))$, where
$\operatorname{Nil}(G):=\\{g\in G:\langle g,x\rangle\text{ is nilpotent for
all }x\in G\\}$ is the hypercentre of $G$.

###  4.1. Connectivity of $\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus 1)]$

Mohammadian and Erfanian [57] obtained the following results analogous to
Theorem 3.2 and Theorem 3.3.

######  Theorem 4.1.

Let $G$ be any group.

  1. (a)

[57, Theorem 2.6-2.7] If $G$ is finite solvable or periodic solvable then the
graph $\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus 1)]$ has at most two
connected components whose diameters are at most $7$.

  2. (b)

[57, Theorem 2.10-2.11] If $G$ is finite or locally finite then
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus 1)]$ has at most six connected
components whose diameters are at most $10$.

Mohammadian and Erfanian [57] also obtained the following characterisations of
supersolvable and solvable groups $G$ such that
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is disconnected.

######  Theorem 4.2.

[57, Theorem 3.2] Let $G$ be a supersolvable group. Then the graph
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is disconnected if and
only if one of the following holds:

  1. (a)

If $G$ is an infinite group, then $G=H\rtimes\langle a\rangle$, where $a\in
G$, $|a|=2$ and $H$ is a subgroup of $G$ on which $a$ acts fixed-point-freely.

  2. (b)

If $G$ is a finite group, then $G=H\rtimes K$ is a Frobenius group with kernel
$H$ and a cyclic complement $K$.

######  Theorem 4.3.

[57, Theorem 3.4] Let $G$ be a finite solvable group with disconnected graph
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus 1)]$. Then there exists a
nilpotent normal subgroup $N$ of $G$ such that one of the following holds:

  1. (a)

$G=N\rtimes H$ is a Frobenius group with the kernel $N$ and a complement $H$.

  2. (b)

$G=(N\rtimes L)\rtimes\langle a\rangle$, where $L$ is a non-trivial cyclic
subgroup of $G$ of odd order which acts fixed-point-freely on $N$, $a\in
N_{G}(L)$ is such that $\langle a\rangle$ acts fixed-point-freely on $L$, and
there exist $a\in N\setminus\\{1\\}$ and $i\in\mathbb{N}$ such that $x^{i}\neq
1$ and $[a,x^{i}]=1$.

Conversely, if either (a) or (b) holds, then
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is disconnected.

Notice that the two situations in Theorem 3.5 and Theorem 4.3, where we get
disconnected $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus 1)]$ and
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus 1)]$, are identical. Therefore,
the following problem arises naturally.

######  Problem 4.1.

Determine whether $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
1)]=\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus 1)]$ if and only if one of
the cases in Theorem 4.3 holds.

###  4.2. Properties of
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus\operatorname{Nil}(G))]$

Mohammadian and Erfanian [57] also considered the subgraph
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus\operatorname{Nil}(G))]$ of
$\mathcal{NCC}(G)$ induced by the set
$\operatorname{Cl}(G\setminus\operatorname{Nil}(G))$ in their study. They
obtain the following characterisations of finite non-nilpotent groups $G$ such
that $\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus\operatorname{Nil}(G))]$ is
empty/triangle-free.

######  Theorem 4.4.

[57, Theorem 4.3] Let $G$ be a finite non-nilpotent group. Then the graph
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus\operatorname{Nil}(G))]$ is an
empty graph if and only if $G\cong S_{3}$.

######  Theorem 4.5.

Let $G$ be a finite non-nilpotent group.

  1. (a)

[57, Theorem 4.8] If $|G|$ is odd then
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus\operatorname{Nil}(G))]$ is
triangle-free if and only if $|G|=21$.

  2. (b)

[57, Theorem 4.9] If $|G|$ is even then
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus\operatorname{Nil}(G))]$ is
triangle-free if and only if $G$ is isomorphic to one of the groups $S_{3}$,
$D_{10}$, $D_{12}$, $A_{4}$, $T_{12}$ or
$\operatorname{\mathop{\mathrm{PSL}}}(2,q)$ where $q\in\\{4,7,9\\}$.

Note that the structure of graph
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus\operatorname{Nil}(G))]$ is not
realized much. If the structures of
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus\operatorname{Nil}(G))]$ for
various families of finite groups are known then one can consider problems
similar to Problems 3.2 – 3.4 for the graph
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus\operatorname{Nil}(G))]$.
Therefore, the following problem is worth mentioning.

######  Problem 4.2.

Determine the structure of
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus\operatorname{Nil}(G))]$ for
various families of finite non-nilpotent groups.

We conclude this section with the following problem analogous to Problem 3.1.

######  Problem 4.3.

Characterize all finite non-nilpotent groups $G$ such that the induced
subgraph
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus\operatorname{Nil}(G))]$ of
$\mathcal{NCC}(G)$ is planar, toroidal, double-toroidal or triple-toroidal.

##  5\. Solvable cojugacy class graph

We write $\mathcal{SCC}(G)$ to denote the SCC-graph of a group $G$. The SCC-
graph of $G$ is a graph whose vertex set is $\operatorname{Cl}(G)$ and two
distinct vertices $a^{G}$ and $b^{G}$ are adjacent if there exist some
elements $x\in a^{G}$ and $y\in b^{G}$ such that $\langle x,y\rangle$ is a
solvable group. In this section, we discuss results on an induced subgraph of
$\mathcal{SCC}(G)$. Bhowal et al. [13, 14] considered the subgraph of
$\mathcal{SCC}(G)$ induced by $\operatorname{Cl}(G\setminus 1)$.

###  5.1. Connectivity of $\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$

Not much is known about the connectivity of
$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$. We have the following
problem whose answer is not known.

######  Problem 5.1.

If $G$ is a finite non-solvable group then determine the number of components
of $\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ and an upper bound for
diameters of its components.

The answers to Problem 5.1 for $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus
1)]$ and $\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus 1)]$ are given in
Theorem 3.3 and Theorem 4.1 respectively. The following results are known
regarding the connectivity of the graph
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus 1)]$.

######  Theorem 5.1.

[13, Theorem 2.1] Let $G$ be a finite group. Then
$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is complete if and only if
$G$ is solvable.

######  Theorem 5.2.

[13, Theorem 2.9] If $G$, $H$ are arbitrary non-trivial groups then the graph
$\mathcal{SCC}(G\times H)[\operatorname{Cl}(G\times H\setminus(1_{G},1_{H}))]$
is connected with diameter $\leq 3$. In particular, $\mathcal{SCC}(G\times
G)[\operatorname{Cl}(G\times G\setminus(1_{G},1_{G}))]$ is connected with
diameter $\leq 3$. Further, $\operatorname{diam}(\mathcal{SCC}(G\times
G)[\operatorname{Cl}(G\times G\setminus(1_{G},1_{G}))])=3$ if and only if
$\operatorname{diam}(\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)])\geq 3$
(possibly infinite).

######  Theorem 5.3.

[13, Theorem 4.4] Let $G$ be a finite group. Let $H$ and $K$ be two subgroups
of $G$ such that $H$ is normal in $G$, $G=HK$ and
$\mathcal{SCC}(H)[\operatorname{Cl}(H\setminus 1)]$,
$\mathcal{SCC}(K)[\operatorname{Cl}(K\setminus 1)]$ are connected. If there
exist two elements $h\in H\setminus\\{1\\}$ and $x\in G\setminus H$ such that
$h^{G}$ and $x^{G}$ are connected in
$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ then
$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is connected.

######  Theorem 5.4.

[13, Theorem 3.3] Let $G$ be a finite group. If $G$ has an element of order
$n=\Pi_{i=1}^{m}p_{i}^{k_{i}}$, where $p_{i}$’s are distinct primes. Then
$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ has a clique of size
$\Pi_{i=1}^{m}(k_{i}+1)-1$.

######  Theorem 5.5.

[13, Theorem 3.5] For any positive integer $d$, there are only finitely many
finite groups $G$ such that the clique number of $\mathcal{SCC}(G)$ is $d$.

######  Problem 5.2.

Do the analogues of Theorem 5.5 hold for the graphs $\mathcal{CCC}(G)$ and
$\mathcal{NCC}(G)$?

We conclude this section with the following result which shows that the graph
$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is triangle-free when
$G\cong\\{1\\},\mathbb{Z}_{2},\mathbb{Z}_{3}$ or $S_{3}$, the symmetric group
of degree $3$.

######  Theorem 5.6.

[13, Theorem 3.4] With the exception of the cyclic groups of orders $1$, $2$
and $3$ and the symmetric group of degree $3$, every finite group $G$ has the
property that $\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ contains a
triangle (that is, has girth $3$).

###  5.2. Genus of $\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$

We have seen various results on genus of the graph
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$ in Subsection 3.4. The
genus of
$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus\operatorname{Sol}(G))]$, where
$\operatorname{Sol}(G):=\\{g\in G:\langle g,x\rangle\text{ is solvable for all
}x\in G\\}$, the solvable radical of $G$, is not studied so far. However, the
following problem is considered in [14].

######  Problem 5.3.

Characterize all finite groups $G$ such that the graph
$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is planar, toroidal,
double-toroidal, triple-toroidal or projective.

Let $N_{k}$ be the connected sum of $k$ projective planes. A simple graph
$\Gamma$ which can be embedded in $N_{k}$ but not in $N_{k-1}$, is called a
graph with crosscap $k$. We write $\bar{\gamma}(\Gamma)$ to denote the
crosscap of $\Gamma$. A graph $\Gamma$ is called projective if
$\bar{\gamma}(\Gamma)=1$. Bhowal et al. [14] obtained the following results
related to Problem 5.3.

######  Theorem 5.7.

[14, Theorem 3.1] Let $G=D_{2n}$. Then

  1. (a)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is planar if and only if
$n=2,3,4,5$ and $7$.

  2. (b)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is toroidal if and only if
$n=6,8,9,10,11$ and $13$.

  3. (c)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is double-toroidal if and
only if $n=12$ and $15$.

  4. (d)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is triple-toroidal if and
only if $n=14$ and $17$.

  5. (e)

$\gamma(\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus
1)])=\begin{cases}\left\lceil\frac{(n-5)(n-7)}{48}\right\rceil,&\text{when
$n\geq 19$ and $n$ is odd}\vspace{.2cm}\\\
\left\lceil\frac{(n-2)(n-4)}{48}\right\rceil,&\text{when $n\geq 16$ and $n$ is
even}.\end{cases}$

  6. (f)

$\bar{\gamma}(\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)])=0$ if and
only if $n=2,3,4,5$ and $7$.

  7. (g)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is projective if and only
if $n=6,8,9$ and $11$.

  8. (h)

$\bar{\gamma}(\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus
1)])=\begin{cases}\left\lceil\frac{(n-5)(n-7)}{24}\right\rceil,&\text{when
$n\geq 13$ and $n$ is odd}\vspace{.2cm}\\\
\left\lceil\frac{(n-2)(n-4)}{24}\right\rceil,&\text{when $n\geq 10$ and $n$ is
even}.\end{cases}$

######  Theorem 5.8.

[14, Theorem 3.2] Let $G=Q_{4m}$. Then

  1. (a)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is planar if and only if
$m=1$ and $2$.

  2. (b)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is toroidal if and only if
$m=3,4$ and $5$.

  3. (c)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is double-toroidal if and
only if $m=6$.

  4. (d)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is triple-toroidal if and
only if $m=7$.

  5. (e)

$\gamma(\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus
1)])=\left\lceil\frac{(m-1)(m-2)}{12}\right\rceil$ for $m\geq 8$.

  6. (f)

$\bar{\gamma}(\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)])=0$ if and
only if $m=1$ and $2$.

  7. (g)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is projective if and only
if $m=3$ and $4$.

  8. (h)

$\bar{\gamma}(\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus
1)])=\begin{cases}3,&\text{when $m=5$}\vspace{.2cm}\\\
\left\lceil\frac{(m-1)(m-2)}{6}\right\rceil,&\text{when $m\geq
6$}.\end{cases}$

######  Theorem 5.9.

[14, Theorem 3.3] Let $G$ be a finite solvable group. Then

  1. (a)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is planar if and only if
$|\operatorname{Cl}(G)|=1,2,3,4$ and $5$.

  2. (b)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is toroidal if and only if
$|\operatorname{Cl}(G)|=6,7$ and $8$.

  3. (c)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is double-toroidal if and
only if $|\operatorname{Cl}(G)|=9$.

  4. (d)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is triple-toroidal if and
only if $|\operatorname{Cl}(G)|=10$.

  5. (e)

$\gamma(\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus
1)])=\left\lceil\frac{(|\operatorname{Cl}(G)|-4)(|\operatorname{Cl}(G)|-5)}{12}\right\rceil$
for $|\operatorname{Cl}(G)|\geq 11$.

  6. (f)

$\bar{\gamma}(\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)])=0$ if and
only if $|\operatorname{Cl}(G)|=1,2,3,4$ and $5$.

  7. (g)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is projective if and only
if $|\operatorname{Cl}(G)|=6$ and $7$.

  8. (h)

$\bar{\gamma}(\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus
1)])=\begin{cases}3,&\text{when $|\operatorname{Cl}(G)|=8$}\vspace{.2cm}\\\
\left\lceil\frac{(|\operatorname{Cl}(G)|-4)(|\operatorname{Cl}(G)|-5)}{6}\right\rceil,&\text{when
$|\operatorname{Cl}(G)|\geq 9$}.\end{cases}$

######  Theorem 5.10.

[14, Theorem 3.4] Let $G=S_{n}$. Then

  1. (a)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is planar if and only if
$n\leq 5$.

  2. (b)

If $n\geq 7$ then $\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is
neither planar, toroidal, double-toroidal nor triple-toroidal.

  3. (c)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is not toroidal if $n=6$.

  4. (d)

If $n\geq 6$ then $\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is not
projective.

######  Theorem 5.11.

[14, Theorem 3.5] Let $G=A_{n}$, the alternating group of degree $n$. Then

  1. (a)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is planar if and only if
$n\leq 6$.

  2. (b)

If $n\geq 9$ then $\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is
neither planar, toroidal, double-toroidal nor triple-toroidal.

  3. (c)

$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is toroidal if and only if
$n=7$.

  4. (d)

If $n\geq 8$ then $\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ is not
projective.

Bhowal et al. [14] also obtained the genus of
$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ when $G$ is certain
projective special linear group.

######  Theorem 5.12.

[14, Theorem 3.6] Let $G=\operatorname{\mathop{\mathrm{PSL}}}(2,q)$, where
$q=2^{d}$ with $d\geq 3$. Then

$\gamma(\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus
1)])=\gamma(K_{q/2})+\gamma(K_{q/2+1})\text{ or
}\gamma(K_{q/2})+\gamma(K_{q/2+1})-1.$

Regarding bounds for genus and crosscap of
$\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ we have the following
results.

######  Theorem 5.13.

[14, Theorem 2.4] Given $n\geq 10$, let $k=p(\lfloor n/2\rfloor)-1$. If
$G=S_{n}$, the symmetric group of degree $n$, and
$\Gamma=\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$ then

$\gamma(\Gamma)\geq\left\lceil\frac{(k-3)(k-4)}{12}\right\rceil\,\,\text{ and
}\,\,\bar{\gamma}(\Gamma)\geq\left\lceil\frac{(k-3)(k-4)}{6}\right\rceil.$

######  Theorem 5.14.

[14, Theorem 2.6] Let $G$ be a finite non-solvable group with non-trivial
centre $Z(G)$. Then

$4\gamma(\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus
1)])\geq(|Z(G)|-3)(|\operatorname{Cl}(G)|-|Z(G)|-2).$

As an application of Theorem 5.14, we have the following bound for $\Pr(G)$
which is the probability that a randomly chosen pair of elements of $G$
commute; also known as commutativity degree of $G$.

######  Theorem 5.15.

[14, Corollary 2.7] Let $G$ be a finite non-solvable group and $|Z(G)|>3$.
Then

$\Pr(G)\leq\frac{4\gamma(\Gamma_{sc}(G))+(|Z(G)|-3)(|Z(G)|+2)}{|G|(|Z(G)|-3)}.$

Many other bounds for $\Pr(G)$ using various group theoretic notions can be
found in [40, 60, 23]. It is worth noting that similar bounds for nilpotency
degree [25] and solvability degree [35] can be obtained by obtaining bounds
for $\gamma(\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus 1)])$ and
$\gamma(\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus 1)])$ similar to Theorem
5.14.

It is intuitive that the order of a finite group $G$ is bounded (above) by a
function of the genus of $\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)]$.
In this regard, we have the following problem.

######  Problem 5.4.

[14, Problem 2.2] Find an explicit bound for the order of a finite group $G$
for which

  1. (a)

$\gamma(\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)])=k$.

  2. (b)

$\bar{\gamma}(\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus 1)])=k$.

We conclude this section noting that problems similar to Problem 5.4 for the
graphs $\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus 1)]$ and
$\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus 1)]$ are worth considering.

##  6\. Concluding remarks

In [22], conditions for holding equalities in the hierarchy (1) were discussed
for various $\mathcal{P}$ graphs of finite groups. For instance, the cyclic
graph of $G$ is equal to the commuting graph of $G$ if and only if $G$
contains no subgroup isomorphic to $\mathbb{Z}_{p}\times\mathbb{Z}_{p}$ (where
$p$ is prime); the commuting graph of $G$ is equal to the nilpotent graph of
$G$ if and only if the Sylow subgroups of $G$ are abelian; the nilpotent graph
of $G$ is equal to the solvable graph of $G$ if and only if $G$ is nilpotent.
It may be interesting to obtain conditions for holding equalities in the
hierarchy (2) for various $\mathcal{PCC}$-graphs.

Let $DV(\mathcal{PCC}(G))$ be the set of all dominant vertices of
$\mathcal{PCC}(G)$. In view of Proposition 2.5,
$DV(\mathcal{CCC}(G))=\operatorname{Cl}(Z(G))$. It is easy to see that
$\operatorname{Cl}(\operatorname{Nil}(G))\subseteq DV(\mathcal{NCC}(G))$ and
$\operatorname{Cl}(\operatorname{Sol}(G))$ $\subseteq DV(\mathcal{SCC}(G))$.
The following problem along with similar problems for $\mathcal{CCC}(G)$ and
$\mathcal{NCC}(G)$ are worth mentioning here.

######  Problem 6.1.

[14, Problem 3.8] Which non-solvable finite groups $G$ have the property that
$|DV(\mathcal{SCC}(G))|=2$?

To make the question of connectedness of $\mathcal{P}$ conjugacy class graphs
interesting it is necessary to determine $DV(\mathcal{PCC}(G))$ (also see
Problem 2.2). Now consider the following problem.

######  Problem 6.2.

Determine whether the induced subgraphs
$\mathcal{CCC}(G)[\operatorname{Cl}(G\setminus Z(G))]$,
$\mathcal{NCC}(G)[\operatorname{Cl}(G)\setminus DV(\mathcal{NCC}(G))]$ and
$\mathcal{SCC}(G)[\operatorname{Cl}(G)\setminus DV(\mathcal{SCC}(G))]$ of
$\mathcal{CCC}(G)$, $\mathcal{NCC}(G)$ and $\mathcal{SCC}(G)$ respectively are
connected. Also, find upper bounds for the diameters of their components.

Note that the induced subgraphs $\mathcal{NCC}(G)[\operatorname{Cl}(G\setminus
FC(G))]$ and $\mathcal{SCC}(G)[\operatorname{Cl}(G\setminus FC(G))]$ of
$\mathcal{NCC}(G)$ and $\mathcal{SCC}(G)$ respectively are not studied yet.
(Recall that $FC(G)$ is the FC-centre of $G$, see Section 3.6.) Therefore,
researchers working in this area may consider these graphs in their study.

The complements of cyclic/commuting/nilpotent/solvable graphs (in other words
non-cyclic/non-commuting/non-nilpotent/non-solvable graphs) of finite groups
were well-studied over the years (see [2, 3, 1, 27, 32, 4, 76, 47, 19]).
However, the complements of $\mathcal{PCC}$-graphs of $G$ are not studied.
Note that $DV(\mathcal{PCC}(G))$ is the set of isolated vertices of the
complement of $\mathcal{PCC}(G)$. Thus, by Proposition 2.5, it follows that
the set of isolated vertices of the complement of $\mathcal{CCC}(G)$ is
$\operatorname{Cl}(Z(G))$. The following problem is equivalent to Problem 2.2.

######  Problem 6.3.

Describe the set of isolated vertices of the complements of NCC- or SCC-graph
of a finite group.

Similarly, we have equivalent problems corresponding to Problems 2.1, 4.1, 4.2
and 6.1 for the complement of $\mathcal{PCC}(G)$. We conclude this paper
noting that problems analogous to Problems 3.1–3.4, 4.3, 5.1–5.4 and 6.2 for
complements of $\mathcal{PCC}$-graphs of $G$ are worth considering.

## Acknowledgments

F. E. Jannat would like to thank DST for the INSPIRE Fellowship (IF200226).

The authors are grateful to Scott Harper for the proof of Theorem 2.6(c).

## References

  * [1] Abdollahi, A. Akbari, S. and Maimani, H. R. Non-commuting graph of a group, J. Algebra, 298(2), 468–492, 2006.
  * [2] Abdollahi, A. and Hassanabadi, A. M. Noncyclic graph of a group, _Commun. Algebra_ 35, 2057–2081, 2007.
  * [3] Abdollahi, A. and Hassanabadi, A. M. Noncyclic graph associated with a group, _J. Algebra Appl._ 8, 243–257, 2009.
  * [4] Abdollahi, A. and Zarrin, M. Non-Nilpotent graph of a group, _Commun. Algebra_ 38(12), 4390–4403, 2010.
  * [5] Abreu, N., Cardoso, D. M., Gutman, I., Martins, E. A. and Robbiano, M. Bounds for the Signless Laplacian energy, _Linear Algebra Appl._ 435(10), 2365–2374, 2011.
  * [6] Afkhami, M., Farrokhi, D. G. M. and Khashyarmanesh, K. Planar, toroidal, and projective commuting and non-commuting graphs, Comm. Algebra 43(7), 2964–2970, 2015.
  * [7] Akbari, B., Lewis, M. L., Mirzajani, J. and Moghaddamfar, A. R. The solubility graph associated with a finite group, Int. J. Algebra Comput. 30(8), 1555–1564, 2020.
  * [8] Alwardi, A., Soner, N. D. and Gutman, I. On the common-neighborhood energy of a graph, Bull. Cl. Sci. Math. Nat. Sci. Math. 36, 49–59, 2011.
  * [9] Arunkumar, G., Cameron, P. J., Nath, R. K. and Selvaganesh, L. Super graphs on groups, I, Graphs Combin. 38, Article No. 100, 2022.
  * [10] Ashrafi, A. R. and Salahshour, M. A. Counting centralizers of a finite group with an application in constructing the commuting conjugacy class graph, Comm. Algebra 51(3), 1105–1116, 2023.
  * [11] Ballester-Bolinches, A., Esteban-Romero, R. and Robinson, D. J. S. On finite minimal non-nilpotent groups, Proc. Amer. Math. Soc., 133, 3455–3462, 2005.
  * [12] Bertram, E. A., Herzog, M., Mann, A. On a graph related to conjugacy classes of groups, Bull. London Math. Soc. 22, 569–575, 1990.
  * [13] Bhowal, P., Cameron, P. J., Nath, R. K. and Sambale B., Solvable conjugacy class graph of groups, Discrete Math. 346(8), 113467, 2023.
  * [14] Bhowal, P., Cameron, P. J., Nath, R. K. and Sambale B., Genus and crosscap of solvable conjugacy class graphs of finite groups, Arch. Math. (accepted for publication).
  * [15] Bhowal, P. and Nath, R. K. Spectral aspects of commuting conjugacy class graph of finite groups, Alg. Struc. Appl. 8(2), 67–118, 2021.
  * [16] Bhowal, P. and Nath, R. K. Genus of commuting conjugacy class graph of certain finite groups, Alg. Struc. Appl. 9(1), 93–108, 2022.
  * [17] Bhowal, P. and Nath, R. K. Spectrum and energies of commuting conjugacy class graph of a finite group, https://arxiv.org/pdf/2003.07142.pdf.
  * [18] Bhowal, P., Nongsiang, D. and Nath, R. K. Solvable graphs of finite groups, Hacet. J. Math. Stat. 49(6), 1955–1964, 2020.
  * [19] Bhowal, P., Nongsiang, D. and Nath, R.K. Non-solvable graphs of groups, _Bull. Malays. Math. Sci. Soc._ 45, 1255–1272, 2022.
  * [20] Brauer, R. and Fowler, K. A. On groups of even order, Ann. of Math. 62, 565–583, 1955.
  * [21] Burness, T. C., Lucchini, A., Nemmi, D. On the soluble graph of a finite group, J. Combin. Theory Ser. A 194, 105708, 2023.
  * [22] Cameron, P. J. Graphs defined on groups, Int. J. Group Theory 11(2), 53–107, 2022.
  * [23] Das, A. K., Nath, R. K. and Pournaki, M. R. A survey on the estimation of commutativity in finite groups, Southeast Asian Bull. Math. 37(2), 161–180, 2013.
  * [24] Dolfi, S., Guralnick, R. M., Marcel, H. and Praeger, C. E. A new solvability criterion for finite groups, J. Lond. Math. Soc. (2) 85, 269–281, 2012.
  * [25] Dubose-Schmidt, H., Galloy, M. D. and Wilson, D. L. Counting nilpotent pairs in finite groups: Some conjectures, Mathematical Sciences Technical Reports (MSTR). 132, 1992. https://scholar.rose-hulman.edu/math_mstr/132.
  * [26] Dutta, P., Bagchi, B. and Nath, R. K. Various energies of commuting graphs of finite nonabelian groups, Khayyam J. Math. 6(1), 27–45, 2020.
  * [27] Dutta, P., Dutta, J. and Nath, R. K. On Laplacian spectrum of non-commuting graph of finite groups, Indian J. Pure Appl. Math. 49(2), 205–216, 2018.
  * [28] Dutta, J. and Nath, R. K. Spectrum of commuting graphs of some classes of finite groups, Matematika 33(1), 87–95, 2017.
  * [29] Dutta, J. and Nath, R. K. Finite groups whose commuting graphs are integral, Mat. Vesnik 69(3), 226–230, 2017.
  * [30] Dutta, J. and Nath, R. K. Laplacian and signless Laplacian spectrum of commuting graphs of finite groups, Khayyam J. Math. 4(1), 77–87, 2018.
  * [31] Dutta, P. and Nath, R. K. Various energies of commuting graphs of some super integral groups, Indian J. Pure Appl. Math. 52(1), 1–10, 2021.
  * [32] Fasfous, W. N. T. and Nath, R. K. Inequalities involving energy and Laplacian energy of non-commuting graphs of finite groups, Indian J. Pure Appl. Math. https://doi.org/10.1007/s13226-023-00519-7.
  * [33] Fasfous, W. N. T., Nath, R. K. and Sharafdini, R. Various spectra and energies of commuting graphs of finite rings, Hacet. J. Math. Stat. 49(6), 1915–1925, 2020.
  * [34] Fasfous, W. N. T., Sharafdini, R. and Nath, R. K. Common neighbourhood spectrum graphs of finite groups, Algebra Discrete Math. 32(1), 33–48, 2021.
  * [35] Fulman, J. E., Galloy, M. D., Sherman, G. J. and Vanderkam, J. M. Counting nilpotent pairs in finite groups, Ars Combin. 54, 161–178, 2000.
  * [36] Gong, S., Li, X., Xu, G., Gutman, I. and Furtula, B. Borderenergetic Graphs, MATCH Commun. Math. Comput. Chem. 74, 321–332, 2015.
  * [37] Parker, C. W. The commuting graph of a soluble group, Bull. Lond. Math. Soc. 45(4), 839–848, 2013.
  * [38] Giudici, M. and Parker, C. W. There is no upper bound for the diameter of the commuting graph of a finite group, J. Combin. Theory, Ser. A 120, 1600–1603, 2013.
  * [39] Guralnick, R. M. and Kantor, W. M. Probabilistic generation of finite simple groups, J. Algebra 234, 743–792, 2000.
  * [40] Guralnick, R. M. and Robinson, G. R. On the commuting probability in finite groups, J. Algebra 300, 509–528, 2006.
  * [41] Gutman, I. The energy of a graph, Ber. Math–Statist. Sekt. Forschungsz. Graz, 103, 1–22, 1978.
  * [42] Gutman, I. Hyperenergetic molecular graphs, J. Serb. Chem. Soc. 64, 199–205, 1999.
  * [43] Gutman, I. Hyperenergetic and Hypoenergetic Graphs, Zb. Rad. (Beogr.) 22, 113–135, 2011.
  * [44] Gutman, I., Abreu, N. M. M., Vinagre, C. T.M., Bonifacioa, A. S and Radenkovic, S. Relation between energy and Laplacian energy, MATCH Commun. Math. Comput. Chem. 59, 343–354, 2009.
  * [45] Gutman, I. and Furtula, B. Graph energies and their applications, Bull. Cl. Sci. Math. Nat. Sci. Math. 44, 29–45, 2019.
  * [46] Gutman, I. and Zhou, B. Laplacian energy of a graph, Linear Algebra Appl. 414, 29–37, 2006.
  * [47] Hai-Reuven, D. Non-solvable graph of a finite group and solvabilizers, https://arxiv.org/pdf/1307.2924.pdf.
  * [48] Herzog, M., Longobardi, M. and Maj, M. On a commuting graph on conjugacy classes of groups, Comm. Algebra 37(10), 3369–3387, 2009.
  * [49] Iranmanesh, A. and Jafarzadeh, A. On the commuting graph associated with the symmetric and alternating groups, J. Algebra Appl. 7, 129–146, 2008.
  * [50] Jannat, F. E. and Nath, R. K. Common neighborhood spectrum and energy of commuting conjugacy class graph, J. Algebr. Syst. 12(2), 301–326, 2025.
  * [51] Jannat, F. E. and Nath, R. K. Common neighborhood (signless) Laplacian spectrum and energy of CCC-graph, https://arxiv.org/pdf/2403.02703.pdf.
  * [52] Jannat, F. E., Nath, R. K. and Das, K. C. Common neighborhood energies and their relations with Zagreb index, https://arxiv.org/pdf/2402.15416.pdf.
  * [53] Landau, E. Über die Klassenzahl der binären quadratischen Formen von negativer Discriminante, Math. Ann. 56, 671–676, 1903.
  * [54] Lewis, M. L. An overview of graphs associated with character degrees and conjugacy class sizes in finite groups, Rocky Mountain J. Math. 38(1), 175–211, 2008.
  * [55] Liu, J. and Liu, B. On the relation between energy and Laplacian energy, MATCH Commun. Math. Comput. Chem. 61, 403–406, 2009.
  * [56] Mohammadian, A., Erfanian, A., Farrokhi, D. G. M. and Wilkens, B. Triangle-free commuting conjugacy class graphs, J. Group Theory 19, 1049–1061, 2016.
  * [57] Mohammadian, A., Erfanian, A., Farrokhi, D. G. M. On the nilpotent conjugacy class graph of groups, Note Mat. 37(2), 77–89, 2017.
  * [58] Morgan, G. L. and Parker, C. W. The diameter of the commuting graph of a finite group with trivial centre, J. Algebra 393, 41–59, 2013.
  * [59] Nath, R. K. Various spectra of commuting graphs of $n$-centralizer finite groups, Int. J. Eng. Sc. Tech. 10(2S), 170–172, 2018.
  * [60] Nath, R. K. and Das, A. K. On a lower bound for commutativity degree, Rend. Circ. Mat. Palermo 59(1), 137–142, 2010.
  * [61] Nath, R. K., Fasfous, W. N. T., Das, K. C. and Shang, Y. Common neighbourhood energy of commuting graphs of finite groups, Symmetry 13(9), 1651–1662, 2021.
  * [62] Neumann, B. H. A problem of Paul Erdős on groups, J. Aust. Math. Soc. 21(4), 467–472, 1976.
  * [63] Nongsiang, D. Double-Toroidal and Triple-Toroidal Commuting Graph, Hacettepe Journal of Mathematics and Statistics, pp. 1–13, 2024, doi:10.15672/hujms.1052964.
  * [64] On-Line Encyclopedia of Integer Sequences, https://oeis.org/
  * [65] Rezaei, M. and Foruzanfar, Z. The commuting conjugacy class graphs of finite groups with a given property, Georgian Math. J. 31(1), 121–127, 2024.
  * [66] Salahshour, M. A. Commuting conjugacy class graph of $G$ when $\frac{G}{Z(G)}\cong D_{2n}$, Math. Interdiscip. Res. 1, 379–385, 2020.
  * [67] Salahshour, M. A. and Ashrafi, A. R. Commuting conjugacy class graphs of finite groups, Alg. Struc. Appl., 7(2), 135–145, 2020.
  * [68] Salahshour, M. A. and Ashrafi, A. R. Commuting conjugacy class graph of finite CA-groups, Khayyam J. Math., 6(1), 108–118, 2020.
  * [69] Schmidt, O. Yu. Groups all of whose subgroups are nilpotent, Mat. Sbornik (Russian), 31, 366–372, 1924.
  * [70] Sharafdini, R., Nath, R. K. and Darbandi, R. Energy of commuting graph of finite AC-groups, Proyecciones, 41(1), 263–273, 2022.
  * [71] Stevanovic, D., Stankovic, I. and Milosevic, M. On the relation between energy and Laplacian energy of graphs, MATCH Commun. Math. Comput. Chem. 61, 395–401, 2009.
  * [72] Tao, Q. and Hou, Y. Q-borderenergetic graphs, AKCE Int. J. Graphs Comb. 17(1), 38–44, 2020.
  * [73] Tura, F. L-borderenergetic graphs, MATCH Commun. Math. Comput. Chem. 77, 37–44, 2017.
  * [74] Walikar, H. B., Ramane, H. S. and Hampiholi, P. R. On the energy of a graph, Graph connections, Eds. R. Balakrishnan, H. M. Mulder and A. Vijayakumar, Allied publishers, New Delhi, 120-123, 1999.
  * [75] White, A. T. Graphs, Groups and Surfaces, North-Holland Mathematics Studies, No. 8., American Elsevier Publishing Co., Inc., New York, 1973.
  * [76] Żak, R. A few remarks on the theory of non-nilpotent graphs, _Commun. Algebra_ 51(11), 4604–4613, 2023.