Common Neighborhood Energy of Commuting Graphs of Finite Groups

: The commuting graph of a ﬁnite non-abelian group G with center Z ( G ) , denoted by Γ c ( G ) , is a simple undirected graph whose vertex set is G \ Z ( G ) , and two distinct vertices x and y are adjacent if and only if xy = yx . Alwardi et al. (Bulletin, 2011, 36, 49-59) deﬁned the common neighborhood matrix CN ( G ) and the common neighborhood energy E cn ( G ) of a simple graph G . A graph G is called CN-hyperenergetic if E cn ( G ) > E cn ( K n ) , where n = | V ( G ) | and K n denotes the complete graph on n vertices. Two graphs G and H with equal number of vertices are called CN-equienergetic if E cn ( G ) = E cn ( H ) . In this paper we compute the common neighborhood energy of Γ c ( G ) for several classes of ﬁnite non-abelian groups, including the class of groups such that the central quotient is isomorphic to group of symmetries of a regular polygon, and conclude that these graphs are not CN-hyperenergetic. We shall also obtain some pairs of ﬁnite non-abelian groups such that their commuting graphs are CN-equienergetic.


Introduction
Let G be a simple graph whose vertex set is V(G) = {v 1 , v 2 , . . . , v n }. The common neighborhood of two distinct vertices v i and v j , denoted by C(v i , v j ), is the set of vertices adjacent to both v i and v j other than v i and v j . The common neighborhood matrix of G, denoted by CN(G), is a matrix of size n whose (i, j)th entry is 0 or |C(v i , v j )| according as i = j or i = j. The common neighborhood matrix is a symmetric matrix, hence all its eigenvalues are real. The common neighborhood eigenvalues are symmetric with respect to the origin for some special class of graphs. There is a nice relation between CN(G) and A(G), the adjacency matrix of G. More precisely, if i = j then the (i, j)th entry of CN(G) is same as the (i, j)th entry of A(G) 2 , which is the number of 2-walks between the vertices v i and v j . Further, the (i, i)th entry of CN(G) is equal to the degree of v i . Hence, CN(G) = A(G) 2 − D(G), where D(G) is the degree matrix of G. Let CN-spec(G) be the spectrum of CN(G). Then CN-spec(G) is the set of all the eigenvalues of CN(G) with multiplicities. If α 1 , α 2 , . . . , α k are the distinct eigenvalues of CN(G) with multiplicities a 1 , a 2 , . . . , a k , respectively, then we write CN-spec(G) = {α a 1 1 , α a 2 2 , . . . , α a k k }. The common neighborhood energy (abbreviated as CN-energy) of the graph G is given by The study of CN-energy of graphs was introduced by Alwardi et al. in [1]. Various properties of CN-energy of a graph can also be found in [1,2]. The motivation of studying E cn (G) comes from the study of E(G), which is well-known as energy of G, a notion introduced by Gutman [3]. Many results on E(G), including some bounds and chemical applications, can be found in [4][5][6][7][8][9][10][11][12][13][14][15]. It is worth recalling that E(G) is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. It is also interesting to note that E(G) can be obtained if E cn (G) is known for some classes of graphs. For instance, E(K n ) = E cn (K n )/(n − 2) and E(K m,n ) = E cn (K m,n ) + 2(n + n), where K n is the complete graph on n vertices and K m,n is the complete bipartite graph on (m + n) It is still an open problem to produce a CN-hyperenergetic graph or to prove the nonexistence of such graph (see [1] (Open problem 1)). In this paper we give an attempt to answer this problem by considering commuting graphs of finite groups.
The commuting graph of a finite non-abelian group G with center Z(G) is a simple undirected graph whose vertex set is G \ Z(G) and two vertices x and y are adjacent if and only if xy = yx. We write Γ c (G) to denote this graph. In [16][17][18][19][20][21][22][23], various aspects of Γ c (G) are studied. In Section 2 of this paper, we derive an expression for computing CN-energy of a particular class of graphs and list a few already known results. In Section 3, we compute CN-energy of commuting graph of certain metacyclic group, dihedral group (which is the group of symmetries of a regular polygon), quasidihedral group, generalized quarternion group, Hanaki group etc. We also consider some generalizations of dihedral group and generalized quarternion group. Two graphs G and H with equal number of vertices are called CN-equienergetic if E cn (G) = E cn (H). In Section 3, we shall also obtain some pairs of finite non-abelian groups such that their commuting graphs are CN-equienergetic. As consequences of our results, in Section 4, we show that Γ c (G) for all G considered in Section 3 are not CN-hyperenergetic. We also identify some positive integers n such that Γ c (G) is not CN-hyperenergetic if G is an n-centralizer group. It is worth mentioning that CN-spectrums of Γ c (G) for certain classes of finite groups have been computed in [24] recently. However, the method adopted here, in computing CN-energy of Γ c (G) for various families of finite groups, is independent of CN-spec(Γ c (G)).
Recall that an n-centralizer group G is a group such that | Cent(G)| = n, where Cent(G) = {C G (w) : w ∈ G} and C G (w) = {v ∈ G : vw = wv} is the centralizer of w (see [25,26]). We also identify some r ∈ Q >0 such that Γ c (G) is not CN-hyperenergetic if Pr(G) = r. Also recall that the commutativity degree of G, denoted by Pr(G), is the probability that a randomly chosen pair of elements of G commute.
Readers may review [27][28][29][30][31][32] for the background and various results regarding this notion. Further, we show that Γ c (G) is not CN-hyperenergetic if Γ c (G) is not planar or toroidal. Note that a graph is planar or toroidal according as its genus is zero or one respectively. Finally, we conclude the paper with a few conjectures.

A Useful Formula and Prerequisites
We write G = G 1 G 2 to denote that G has two components namely G 1 and G 2 . Also, lK m denotes the disjoint union of l copies of the complete graph K m on m vertices. We begin this section with the following two key results of Alwardi et al. [1].
Lemma 1 ([1] Example 2.1). If K n denotes the complete graph on n vertices then E cn (K n ) = 2(n − 1)(n − 2). Now we derive a formula for CN-energy of graphs which are disjoint unions of some complete graphs. The following theorem is very useful in order to compute CN-energy of commuting graphs of finite groups. Theorem 2. Let G = l 1 K m 1 l 2 K m 2 · · · l k K m k , where l i K m i denotes the disjoint union of l i copies of the complete graphs K m i on m i vertices for 1 ≤ i ≤ k. Then Proof. By Theorem 1 we have Therefore, the result follows from Lemma 1.
The Hanaki group A(n, p) of order p 3n given by under matrix multiplication where F = GF(p n ) and p is a prime, then Γ c (G) = (p n + 1)K p 2n −p n .

CN-Energy of Commuting Graphs
In this section, we compute E cn (Γ c (G)) for several classes of finite non-abelian groups.
Theorem 3. Let G be a finite group with center Z(G). If G Z(G) is isomorphic to 1. The Suzuki group Sz (2), then Proof. By Lemma 2 and Theorem 2 we have Hence, the result follows on simplification.
We have the following two corollaries of Theorem 3.

Corollary 1.
Let G be isomorphic to one of the following groups 1.
Proof. If G is isomorphic to one of the above listed group then it is of order 16. Therefore, |Z(G)| = 4 and so G Z(G) Hence, putting p = 2 in Theorem 3 (2) we get the required result.

Corollary 2.
Let G be a non-abelian group.

1.
If G is of order p 3 , for any prime p, then Hence the result follows from Theorem 3 (2).
In the following theorems we compute E cn (Γ c (G)) for more families of groups.
Theorem 4. Let G be a non-abelian group.
1. If G is of order pq, where p and q are primes with p | (q − 1), then Proof.
(1) If G is of order pq then, by Lemma 3 (1) and Theorem 2, we have This gives the required result on simplification.
(3) By Lemma 3 (3) and Theorem 2 we have which gives the required result.
(4) By Lemma 3 (4) and Theorem 2 we have which gives the required result on simplification.
Theorem 5. Let G be a non-abelian group.

2.
If G is the Hanaki group A(n, p) then Proof. The result follows from Lemma 4 and Theorem 2.
Note that all the groups considered above are abelian centralizer group (in short, AC-group). Now we present a result on E cn (Γ c (G)) if G is a finite AC-group.
Hence, H × K is an AC-group and so, by Theorem 6, the result follows.
We shall conclude this section by obtaining some pairs of finite non-abelian groups such that their commuting graphs are CN-equienergetic. Proposition 1. The commuting graphs of D 4k and Q 4k for k ≥ 2 are CN-equienergetic.

Proposition 2.
The commuting graphs of D 2 k , Q 2 k and QD 2 k for k ≥ 4 are pairwise CNequienergetic.

Some Consequences
In this section we derive some consequences of the results obtained in Section 3.

Theorem 7.
Let G be a finite group with center Z(G). If G Z(G) is isomorphic to Sz(2), Z p × Z p or D 2m (where p is any prime and m ≥ 2) then Γ c (G) is not CN-hyperenergetic. Since |V(Γ c (G))| = 19|Z(G)|, by Lemma 1 we have
We have the following two corollaries.

Corollary 4.
If G is isomorphic to one of the groups listed in Corollary 1, then Γ c (G) is not CN-hyperenergetic.
Proof. Since G Z(G) is isomorphic to Z 2 × Z 2 , the result follows from Theorem 7 considering p = 2.

Corollary 5.
Let G be a non-abelian group. If G is isomorphic to M 2mn , D 2m , Q 4n or a group of order p 3 then Γ c (G) is not CN-hyperenergetic.
Proof. If G is isomorphic to M 2mn , D 2m or Q 4n then G Z(G) is isomorphic to some dihedral groups. If G is isomorphic to a group of order p 3 then G Z(G) is isomorphic to Z p × Z p . Hence, by Theorem 7, the result follows.
We have the following results regarding commuting graphs of finite n-centralizer groups.
As a corollary to Theorems 8 and 10 we have the following result. Corollary 6. Let G be a finite non-abelian group and {x 1 , x 2 , . . . , x r } be a set of pairwise noncommuting elements of G having maximal size. Then Γ c (G) is not CN-hyperenergetic if r = 3, 4.
Proof. By [34] (Lemma 2.4), we have that G is a 4-centralizer or a 5-centralizer group according as r = 3 or 4. Hence the result follows from Theorems 8 and 10.
Theorem 11. Let G be a non-abelian group. If G is isomorphic to QD 2 n , PSL(2, 2 k ), A(n, σ), GL(2, q), A(n, p) or a group of order pr, where p and r are primes with p | (r − 1) and q = p m > 2, then Γ c (G) is not CN-hyperenergetic.
It is already mentioned that Pr(G), the commutativity degree of a group G, is the probability that a randomly chosen pair of elements of G commute. Therefore, it measures the abelianness of a group. For any finite group G, its commutativity degree can be computed using the formula where k(G) is the number of conjugacy classes in G. In finite group theory, it is an interesting problem to find all the rational numbers r ∈ (0, 1] such that Pr(G) = r for some finite group G. Over the decades, many values of such r have obtained and characterized finite groups such that Pr(G) = r. In the following theorem we list some values of r such that Γ c (G) is not CN-hyperenergetic if Pr(G) = r.  [35] (p. 246) and [36] (p. 451)). Hence, the result follows from Theorem 7. Proof. We have G ∼ = A 5 × K for some abelian group K, by [27] (Proposition 3.3.7). It can be seen that Γ c (G) = 5K 3|K| 10K 2|K| 6K 4|K| . Therefore, by Theorem 2, we have Additionally, by Lemma 1, we have E cn (K 59|K| ) = 2(3481|K| 2 − 177|K| + 2). Therefore This completes the proof.
The following three theorems show that Γ c (G) is not CN-hyperenergetic if Γ c (G) is planar/toroidal or the complement of Γ c (G) is planar.
Theorem 16. Let G be a finite non-abelian group. If Γ c (G) is toroidal then Γ c (G) is not CNhyperenergetic.
We also have the following result.
Theorem 17. Let G be a finite non-abelian group. If the complement of Γ c (G) is planar then Γ c (G) is not CN-hyperenergetic.
In view of the above results we conclude this paper with a few conjectures.

Conjecture 1.
A planar or toroidal graph is not CN-hyperenergetic.

Conjecture 2. Γ c (G) is not CN-hyperenergetic.
Conjecture 3. If G = l 1 K m 1 l 2 K m 2 · · · l k K m k , where l i K m i denotes the disjoint union of l i copies of the complete graphs K m i on m i vertices for 1 ≤ i ≤ k, then it is not CN-hyperenergetic.