A survey on conjugacy class graphs of groups

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.


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 x, y is abelian.Using other group types such as cyclic, nilpotent, solvable, . . .graphs have been defined on groups.Given a group type P (for instance, cyclic, abelian, nilpotent, solvable etc.), we define a graph on a group G, called the P graph of G, whose vertex set is G and two distinct vertices x and y are adjacent if x, y is a P group.In this nomenclature 'abelian graph' is nothing but the commuting graph.These graphs forms the following hierarchy (where A ⊆ B denotes that A is a spanning subgraph of B): Cyclic graph ⊆ Commuting graph ⊆ Nilpotent graph ⊆ 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 P(G) = {g ∈ G : g, h is a P group for all h ∈ G}.Then P(G) is the set of all dominant vertices of P graph of G.
In the four cases just described, 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 P graph induced by G \ P(G) is an interesting problem (see [49,38,37,58,18,7,21]).Of course, this problem is trivial unless 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 P conjugacy class graph of G, or for short the PCC-graph.The 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 ∈ a G and y ∈ b G such that x, y is a P group.We have the following hierarchy in case of P conjugacy class graph: (2) Cyclic CC-graph ⊆ CCC-graph ⊆ NCC-graph ⊆ 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 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 P graph is a "blow-up" of the PCC-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.

General remarks and examples
We begin with a general observation about conjugacy class graphs which is often useful.Let P be any group-theoretic property, and let Γ be the PCC-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 ∈ C i for i = 1, 2 such that the group g 1 , g 2 has property P. In fact a stronger condition holds: if {C 1 , C 2 } is an edge, then for any h 1 ∈ C 1 there exists h 2 ∈ C 2 such that h 1 , h 2 has property P. For let g 1 , g 2 be as in the definition.There exists x ∈ G such that g and since 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 Γ, there are only finitely many finite groups G whose commuting, nilpotent or solvable conjugacy class graph is isomorphic to Γ.
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? Figure 1 shows the conjugacy classes of the symmetric group S 4 , with the CCCand 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 NCCgraph.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 pelements.Let C 1 and C 2 be two such classes, and take g i ∈ C i ∩ P .Then g 1 , g 2 ≤ 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 ∈ C 1 .By our general remark about conjugacy class graphs, there exist h i ∈ C i for all i such that h i commutes with h = h 1 .Then h 1 , . . ., h r = G, and so h ∈ 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 ∈ G, there exists x ∈ G such that g, h x 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 ∈ G where g is a p-element and h a q-element, there exists x ∈ 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 ∈ G such that g, h x 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 PSL(2, 2 a ) for a ≥ 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 g, h 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 ∈ G, there exists x ∈ 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 ∈ 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 ′ and h ′ of g and h respectively so that g ′ , h ′ 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 P be one of the properties "commutative", "nilpotent", or "solvable".A necessary and sufficient condition for the P graph and the conjugacy super P graph to be equal is as follows.
(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 ∈ Q with y x = y, then (x −1 ) y x = [y, x] = y −1 y x is a non-identity q-element.But x, x is nilpotent, so by assumption x y , x 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 ∈ G such that for all nontrivial x ∈ G there exists g ∈ G such that x, s g = 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 P supergraph, and therefore are joined in the 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 ∈ M such that M/N = Ng, Ng y .In particular, Ng, Ng y is nonsolvable and hence g, g y is non-solvable.However, g, g = g 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.

Commuting conjugacy class graph
We write CCC(G) to denote the CCC-graph of a group G.The CCC-graph of G is a graph whose vertex set is Cl(G) := {x G : x ∈ 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 ∈ a G and y ∈ b G such that x, y is an abelian group.In this section, we discuss results on various induced subgraphs of CCC(G).Herzog et al. [48] considered three induced graphs of CCC(G) induced by Cl(G \ 1), Cl(G \ Z(G)) and Cl(G \ F C(G)), where Cl(S) = {x G : x ∈ S} for any subset S of G, and ]. 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 CCC(G)[Cl(G \ 1)] is complete.
where S is a non-trivial cyclic subgroup of G of odd order which acts fixed-point-freely on H, x ∈ N G (S) is such that x acts fixed-pointfreely on S, and there exist In the following theorem Herzog et al. [48] determined all periodic groups G such that CCC(G)[Cl(G \ Z(G))] is empty.Theorem 3.6.[48, Theorem 19] Let G be a periodic non-abelian group.Then In 2016, Mohammadian et al. [56] classified all finite groups G such that the graph CCC(G)[Cl(G \ Z(G))] is triangle-free and obtained the following results.
] is triangle-free, then G is isomorphic to one of the groups P SL(2, q) (q ∈ {4, 7, 9}), P SL (3,4) 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 G Z(G) ∼ = Z p × Z p or D 2n , where p is a prime and As a corollary to Theorem 3.12, we get the structure of 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 Salahshour and Ashrafi [67] determined the structures of CCC(G)[Cl(G \ Z(G))] when G is a finite non-abelian group such that G Z(G) has order p 2 or p 3 as given in the following theorems.
where p is a prime.In a recent work, Rezaei and Foruzanfar [65] have determined the structure of CCC(G)[Cl(G \ Z(G))] when G Z(G) is isomorphic to a Frobenius group of order pq or p 2 q, where p, q are primes.

Genus of CCC(G)[Cl(G \ Z(G))
].For any graph Γ, we write γ(Γ) to denote its genus.The genus of Γ is the smallest integer k ≥ 0 such that Γ can be embedded on the surface obtained by attaching k handles to a sphere.If γ(Γ) is equal to 0, 1, 2, or 3, then Γ is called planar, toroidal, double-toroidal, or triple-toroidal, respectively.Clearly, γ(K 1 ) = γ(K 2 ) = 0.For n ≥ 3, by [75,, we have where ⌈a⌉ 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 This problem was considered by Bhowal and Nath [15] and they characterised the dihedral groups, generalized quaternion groups and semidihedral groups such that CCC(G)[Cl(G \ Z(G))] is planar, toroidal, double-toroidal or triple-toroidal.We have the following theorems for instance.
It may be interesting to continue similar study for the groups with known/unknown structures of CCC(G)[Cl(G \ Z(G))] and answer Problem 3.1.

Various spectra and energies of CCC(G)[Cl(G \ Z(G))
].The spectrum of a finite graph Γ with vertex set V (Γ), denoted by Spec(Γ), is the set of eigenvalues of its adjacency matrix with multiplicities.If Spec(Γ) = {α a 1 1 , α a 2 2 , . . ., α a k k } for some Γ then we mean that α 1 , α 2 , . . ., α k are the eigenvalues of the adjacency matrix of Γ with multiplicities a 1 , a 2 , . . ., a k respectively.Similarly, L-spec(Γ) and Q-spec(Γ) denote the Laplacian spectrum (L-spectrum) and signless Laplacian spectrum (Q-spectrum) of Γ i.e., the set of eigenvalues of the Laplacian and signless Laplacian matrices of Γ respectively.A graph Γ is called integral/L-integral/Q-integral if Spec(Γ)/L-spec(Γ)/Q-spec(Γ) contains only integers.To determine all the integral/ L-integral/Q-integral graphs is a general problem in graph theory.Various spectra of C(G)[G \ Z(G)] were computed in [28,29,30,59] and obtained various groups where D(Γ) is the degree matrix of Γ and tr(D(Γ)) is the trace of D(Γ).In 1978, Gutman [41] introduced the notion of energy of a graph.In Huckel theory, π-electron energy of a conjugated carbon molecule is approximated by E(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 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 C(G)[G \ Z(G)] is considered in [26,31].

Conjecture 3.20. Any finite graph
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 C(G)[G \ Z(G)] was considered.
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.

Comparing various energies of CCC(G)[Cl(G \ Z(G))
] they also obtained the following results.
We conclude this section noting that problems analogous to Problems 3.2 -3.4 for various common neighborhood spectrum and energies of CCC(G)[Cl(G \ 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 C(G)[G \ Z(G)] were considered in [33,34,61] 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 ∼ = S 3 and G has the structure described in Theorem 3.28.

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

Solvable cojugacy class graph
We write SCC(G) to denote the SCC-graph of a group G.The SCC-graph of G is a graph whose vertex set is Cl(G) and two distinct vertices a G and b G are adjacent if there exist some elements x ∈ a G and y ∈ b G such that x, y is a solvable group.In this section, we discuss results on an induced subgraph of SCC(G).Bhowal et al. [13,14] considered the subgraph of SCC(G) induced by Cl(G \ 1).Theorem 5.14.[14, Theorem 2.6] Let G be a finite non-solvable group with nontrivial centre Z(G).Then 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.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]  We conclude this section noting that problems similar to Problem 5.4 for the graphs CCC(G)[Cl(G \ 1)] and N CC(G)[Cl(G \ 1)] are worth considering.

Concluding remarks
In [22], conditions for holding equalities in the hierarchy (1) were discussed for various 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 Z p × 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 PCC-graphs.
Let DV (PCC(G)) be the set of all dominant vertices of PCC(G).In view of Proposition 2.5, DV (CCC(G)) = Cl(Z(G)).It is easy to see that Cl(Nil(G)) ⊆ DV (N CC(G)) and Cl(Sol(G)) ⊆ DV (SCC(G)).The following problem along with similar problems for CCC(G) and N CC(G) are worth mentioning here.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 PCC-graphs of G are not studied.Note that DV (PCC(G)) is the set of isolated vertices of the complement of PCC(G).Thus, by Proposition 2.5, it follows that the set of isolated vertices of the complement of CCC(G) is 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 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 PCC-graphs of G are worth considering.

Theorem 3 . 1 .
[48,  Proposition 1]  Let G be a residually X-group.Then the graph CCC(G)[Cl(G \ 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 CCC(G)[Cl(G \ 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 CCC(G)[Cl(G \ 1)] has at most two connected components, each of diameter ≤ 9. Theorem 3.3.[48, Theorems 13-14] Let G be a finite group or a locally finite group.Then CCC(G)[Cl(G\1)] has at most six connected components, each of diameter ≤ 19.The following theorems give characterisation of supersolvable/solvable groups such that CCC(G)[Cl(G \ 1)] is disconnected.Theorem 3.4.[48, Proposition 7] Let G be a supersolvable group.Then the graph CCC(G)[Cl(G \ 1)] is disconnected if and only if G is either of the groups given in the following two types: (a) G = A ⋊ x , where x ∈ G, |x| = 2 and A is a subgroup of G on which x acts fixed-point-freely.(b) G is finite and G = A ⋊ 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 CCC(G)[Cl(G \ 1)] is disconnected.Then there exists a nilpotent normal subgroup H of G such that one of the following holds: (a) G = H ⋊ T is a Frobenius group with the kernel H and a complement T .(

Theorem 3 . 7 .
[56, Theorem 2.3]  If G is a finite group of odd order and the graph CCC(G)[Cl(G \ 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 CCC is the centralizer of a ∈ G. Consider the equivalence relation ∼ on Cent(G) \ {G} given by C G (a) ∼ 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).

Theorem 3 ..
14. [67, Theorem 3.1] Let G be a finite non-abelian group with centre Z(G) and G Z(G) ∼ = Z p ×Z p , where p is prime.Then CCC(G)[Cl(G\Z(G))] = (p+1)K n , where n = (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 | G Z(G) | = p 3 , where p is a prime.Then one of the following is satisfied: group of order p 4 .Ashrafi and Salahshour [10, Theorem 1.2] also obtained the structure of CCC

(
that the following problem is still open.Problem 3.2.Determine all the finite non-abelian groupsG such that C(G)[G\Z(G)] is integral/L-integral/Q-integral.The energy, Laplacian energy (L-energy) and signless Laplacian energy (Q-energy) of Γ denoted by E(Γ), LE(Γ) and LE + (Γ) respectively are given by , Bhowal and Nath considered problems corresponding to the Problems 3.2-3.4for CCC(G)[Cl(G \ 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

3. 6 .
Properties of CCC(G)[Cl(G\F C(G))].The FC-centre of a group G is the set of elements x ∈ G such that x G is finite.Herzog et al.[48] obtained the following resultsfor the induced subgraph CCC(G)[Cl(G \ F C(G))] of CCC(G) induced by Cl(G) \ {g G : g ∈ F C(G)} when G is a periodic group.Theorem 3.28.[48, Theorem 22] Let G be a periodic group such that the graph CCC(G)[Cl(G \ F C(G))] is empty.Write F = F C(G) and suppose that there exists xF ∈ G/F such that |xF | = 3.Then G has the following structure: G = F ⋊ a, b , where |a| = 3, |b| = 2, a b = a −1 (i.e.a, b ∼ = S 3 ), F is an elementary abelian 2-group, F = D 1 × D 2 , where D 1 = {[d, b] : d ∈ F }, D 2 = {[d, ab] : d ∈ 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 = F C(G) and the graph CCC(G)[Cl(G \ F C(G))] is empty.Theorem 3.29.[48, Theorem 23] Let G be a periodic group such that the graph CCC

4. 1 .
Connectivity of N CC(G)[Cl(G\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.(a) [57, Theorem 2.6-2.7]If G is finite solvable or periodic solvable then the graph N CC(G)[Cl(G \ 1)] has at most two connected components whose diameters are at most 7. (b) [57, Theorem 2.10-2.11]If G is finite or locally finite then N CC(G)[Cl(G \ 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 N CC(G)[Cl(G \ 1)] is disconnected.Theorem 4.2.[57, Theorem 3.2] Let G be a supersolvable group.Then the graph N CC(G)[Cl(G \ 1)] is disconnected if and only if one of the following holds:

( a )
If G is an infinite group, then G = H ⋊ a , where a ∈ G, |a| = 2 and H is a subgroup of G on which a acts fixed-point-freely.(b) If G is a finite group, then G = H ⋊ 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 N CC(G)[Cl(G \ 1)].Then there exists a nilpotent normal subgroup N of G such that one of the following holds: (a) G = N ⋊ H is a Frobenius group with the kernel N and a complement H.(b) G = (N ⋊ L) ⋊ a , where L is a non-trivial cyclic subgroup of G of odd order which acts fixed-point-freely on N, a ∈ N G (L) is such that a acts fixed-pointfreely on L, and there exist a ∈ N \ {1} and i ∈ N such that x i = 1 and [a, x i ] = 1.Conversely, if either (a) or (b) holds, then N CC(G)[Cl(G \ 1)] is disconnected.Notice that the two situations in Theorem 3.5 and Theorem 4.3, where we get disconnected CCC(G)[Cl(G \ 1)] and N CC(G)[Cl(G \ 1)], are identical.Therefore, the following problem arises naturally.

Problem 4 . 1 . 4 . 2 .
Determine whether CCC(G)[Cl(G \ 1)] = N CC(G)[Cl(G \ 1)] if and only if one of the cases in Theorem 4.3 holds.Properties of N CC(G)[Cl(G \ Nil(G))].Mohammadian and Erfanian [57] also considered the subgraph N CC(G)[Cl(G \ Nil(G))] of N CC(G) induced by the set Cl(G \ Nil(G)) in their study.They obtain the following characterisations of finite non-nilpotent groups G such that N CC(G)[Cl(G \ Nil(G))] is empty/triangle-free.Theorem 4.4.[57, Theorem 4.3] Let G be a finite non-nilpotent group.Then the graph N CC(G)[Cl(G \ Nil(G))] is an empty graph if and only if G ∼ = S 3 .Theorem 4.5.Let G be a finite non-nilpotent group.(a) [57, Theorem 4.8] If |G| is odd then N CC(G)[Cl(G \ Nil(G))] is triangle-free if and only if |G| = 21.(b) [57, Theorem 4.9] If |G| is even then N CC(G)[Cl(G \ 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 PSL(2, q) where q ∈ {4, 7, 9}.Note that the structure of graph N CC(G)[Cl(G \ Nil(G))] is not realized much.If the structures of N CC(G)[Cl(G \ 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 N CC(G)[Cl(G \ Nil(G))].Therefore, the following problem is worth mentioning.

Problem 4 . 2 .
Determine the structure of N CC(G)[Cl(G\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 N CC(G)[Cl(G \ Nil(G))] of N CC(G) is planar, toroidal, double-toroidal or triple-toroidal.

5. 1 .
Connectivity of SCC(G)[Cl(G \ 1)].Not much is known about the connectivity of SCC(G)[Cl(G \ 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 SCC(G)[Cl(G\1)] and an upper bound for diameters of its components.The answers to Problem 5.1 for CCC(G)[Cl(G \ 1)] and N CC(G)[Cl(G \ 1)] are given in Theorem 3.3 and Theorem 4.1 respectively.The following results are known regarding the connectivity of the graph CCC(G)[Cl(G \ 1)].Theorem 5.1.[13, Theorem 2.1] Let G be a finite group.Then SCC(G)[Cl(G \ 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 SCC