On the energy of non-commuting graphs

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Journal of Linear Topological Algebra Vol 06 No 17 135-146 On the energy of non-commuting graphs M Ghorbani a Z Gharavi-Alkhansari a a Department of Mathematics Faculty of Science Shahid Rajaee

1 Journal of Linear Topological Algebra Vol 06 No On the energy of non-commuting graphs M Ghorbani a Z Gharavi-Alkhansari a a Department of Mathematics Faculty of Science Shahid Rajaee Teacher Training University Tehran Iran Received 1 April 017; Revised 8 July 017; Accepted 9 August 017 Abstract For given non-abelian group G the non-commuting (NC)-graph Γ(G) is a graph with the vertex set G \ Z(G) two distinct vertices x y V (Γ) are adjacent whenever xy yx The aim of this paper is to compute the spectra of some well-known NC-graphs c 017 IAUCTB All rights reserved Keywords: Non-commuting graph characteristic polynomial linear group 010 AMS Subject Classification: E46 53C35 57S0 1 Introduction Paul Erdős was the first person who considered the non-commuting graph to answer a question on the size of the cliques of a graph in 1975 see [7] In general the noncommuting graph or briefly the N C graph Γ(G) associated to the non-abelian group G with center Z(G) is a simple undirected graph with the vertex set G \ Z(G) in which two vertices join whenever they don t commute For background materials about NC graphs we encourage the interested reader to see references [1 5 6] For given graph Γ its characteristic polynomial is defined as χ(γ λ) = det(λi A) where A is the adjacency matrix of Γ The eigenvalues of Γ are the roots of this polynomial the multi-set {λ 1 λ n } of eigenvalues of A forms the spectrum of Γ By this notation the energy of Γ is defined as [4]: E(Γ) = n λ i i=1 Corresponding author address: mghorbani@srttuedu (M Ghorbani) Print ISSN: c 017 IAUCTB All rights reserved Online ISSN:

2 136 M Ghorbani et al / J Linear Topological Algebra 06(0) (017) Here in Section we compute the spectrum of NC graph of group G where G {QD n P SL( k ) GL( q)} in Section 3 at first we compute the related Laplacian eigenvalues then we compute the energy of these graphs Main Results Here we find the spectrum of NC-graph of three following groups: the projective special linear group P SL( k ) where k the general linear group GL( q) where q = p n (p is a prime integer n 4) the quasi-dihedral group QD n with the following presentations: QD n = a b : a n 1 = b = 1 bab 1 = a n 1 Theorem 1 The spectrum of NC- graph of group QD n is {[ n 1 k] 1 [ ] n 1 [0] 3 n 3 [ n 1 + k] 1 } where k = (5 n 1)( n 1) Proof The group QD n is a non-abelian group of order n Z(QD n) = {1 a n } For non-central elements the centralizers are as follows C QD n (a) = C QD n (a i ) = a for 1 i n 1 1 i n C QD n (a j b) = {1 a n a j b a j+n b} for 1 j n Since the centralizers are abelian subgroups QD n is an AC-group therefore Γ(QD n) is a multipartite graph with parts V i s (0 i n ) as follows: V 0 = {a a a n 1 a n +1 a n 1 1 } V j = {a j b a j+n b} 1 j n By putting p = n 1 q = p/ the adjacency matrix of Γ(QD n) is M = ( ) 0q J q (q+1) J J (q+1) q (J I) = A J q+1 The spectrum of J is {[0] 1 [] 1 } by [3 Lemma ] the spectrum of A is {[0] q 1 [ 1] q [ q q(5(q + 1) 1) Now by using [3 Theorem 1] the proof is completed ] 1 [ q + q(5(q + 1) 1) ] 1 } Theorem The spectrum of NC-graph of the projective special linear group P SL( k ) where k is {[ u] t 1 [ u + 1] u [ u + ] s 1 [0] (u+1)(u )+s(u 3)+t(u 1) [x 1 ] 1 [x ] 1 [x 3 ] 1 }

3 M Ghorbani et al / J Linear Topological Algebra 06(0) (017) where u = k s = k 1 ( k + 1) t = k 1 ( k 1) x 1 x x 3 are the roots of the equation x 3 +( k+ 3k )x +( 3k+1 4k+1 +5 k ( k 1))x+ 4k+1 + 3k 5k k+1 Proof The center of non-abelian group P SL( k ) of order k ( k 1) is trivial By [1 Proposition 31] for non-central elements of P SL( k ) the set of centralizers is {gp g 1 gag 1 gbg 1 : g P SL( k )} where P is an elementary abelian -group of order k A B are cyclic subgroups of order k 1 k + 1 respectively Let u = k s = k 1 ( k + 1) t = k 1 ( k 1) The number of conjugates of P A B in P SL( k ) are u + 1 s t respectively All centralizers of P SL( k ) are abelian since it is an AC-group Also it is not difficult to see that they are disjoint from each other This yields Γ(P SL( k )) is a multipartite graph Let p = u 1 q = u then the adjacency matrix of Γ(P SL( k )) is the following block matrix: 0 p J p J p J p q J p q J p u J p u J p J p J p 0 p J p q J p q J p u J p u J M = q p J q p 0 q J q J q u J q u J q p J q p J q 0 q J q u J q u J u p J u p J u q J u q 0 u J u J u p J u p J u q J u q J u 0 u (u+1)+s+t Let N r = xi + rj then det(xi M) = A B C D = det(a)det(d CA 1 B) where N 1 p 0 0 Np 1 0 A = N 1 p 0 Nq 1 0 Nq 1 0 Nq 1 (u+1)+s

4 138 M Ghorbani et al / J Linear Topological Algebra 06(0) (017) N 1 p u B = Nq u 1 ((u+1)+s) t (u + 1)J u p J u p J u p sj u q J u q J u q C = t (u+s+1) D = J u J u 0 Nu 1 0 Nu 1 Nu t+1 t We have so det(a) = det(n 1 p ) u+1 det(n 1 q ) s = x (u+1)(u )+s(u 3) (x + (u 1)) u+1 (x + (u )) s det(d CA 1 B) = (x x 1)(x x )(x x 3 ) (x + (u 1))(x + (u )) xt(u 1) (x + u) t 1 where x 1 x x 3 are the roots of the following equation x 3 + ( k+ 3k )x + ( 3k+1 4k k ( k 1))x + 4k+1 + 3k 5k k+1 Thus det(xi M) = det(a)det(d CA 1 B) Hence the result follows = x (u+1)(u )+s(u 3)+t(u 1) (x + (u 1)) u (x + (u )) s 1 (x + u) t 1 (x x 1 )(x x )(x x 3 ) Theorem 3 The spectrum of NC- graph of the general linear group GL( q) where q = p n > p is prime integer is given by {[ s] s 1 [ k] t 1 [ l] u 1 [0] l(u 1)+s(s 1)+t(k 1) [x 1 ] 1 [x ] 1 [x 3 ] 1 }

5 M Ghorbani et al / J Linear Topological Algebra 06(0) (017) where u = q(q+1) s = q(q 1) t = q + 1 l = (q 1)(q ) k = (q 1) x 1 x x 3 are the roots of the equation x 3 + ( q 4 + q 3 + 4q + )x + ( q 6 + 6q 5 q 4 13q q 5q)x q 8 + 5q 7 8q 6 + q 5 + 7q 4 7q 3 + q Proof The order of non-abelian group GL( q) is (q 1)(q q) it is well-known that Z(GL( q)) = q 1 By [1 Proposition 36] all non-central elements of GL( q) have the set of centralizers as {gdg 1 gig 1 gp Z(GL( q))g 1 : g GL( q)} where D is the subgroup of GL( q) consisting of all diagonal matrices I is a cyclic subgroup GL( q) having order q 1 P is the Sylow p-subgroup of GL( q) consisting of all upper triangular matrices whose diagonal entries are 1 The orders of D P Z(GL( q)) are (q 1) q(q 1) respectively The number of conjugates of D I P Z(GL( q)) in GL( q) are q(q+1) q(q 1) q + 1 respectively Let u = q(q+1) s = q(q 1) t = q+1 l = (q 1)(q ) k = (q 1) Similar to the last case Γ(GL( q)) is a multipartite graph the order of its parts are D Z(GL( q)) = l I Z(GL( q)) = s P Z(GL( q)) Z(GL( q)) = k The adjacency matrix of Γ(GL( q)) has the following form: 0 l J l J l J l k J l k J l s J l s J l J l J l 0 l J l k J l k J l s J l s J M = k l J k l 0 k J k J k s J k s J k l J k l J k 0 k J k s J k s J s l J s l J s k J s k 0 s J s J s l J s l J s k J s k J s 0 s u+t+s Let N r = xi + rj similar to the proof of last theorem we have det(xi M) = det(a)det(d CA 1 B) where N 1 l 0 0 N l 1 0 N A = 1 l 0 Nk 1 0 Nk 1 0 Nk 1 u+t

6 140 M Ghorbani et al / J Linear Topological Algebra 06(0) (017) N 1 l s B = Nk s 1 (u+t) s uj s l J s l J s l tj s k J s k J s k C = s (u+t) D = J s J s 0 Ns 1 0 Ns 1 Ns s+1 s We have det(a) = det(n 1 l )u det(n 1 k )t = x u(l 1)+t(k 1) (x + l) u (x + k) t hence det(d CA 1 B) = (x x 1)(x x )(x x 3 ) x s(s 1) (x + s) s 1 (x + l)(x + k) where x 1 x x 3 are the roots of the equation Thus x 3 + ( q 4 + q 3 + 4q + )x + ( q 6 + 6q 5 q 4 13q q 5q)x q 8 + 5q 7 8q 6 + q 5 + 7q 4 7q 3 + q det(xi M) = x l(u 1)+s(s 1)+t(k 1) (x + l) u 1 (x + k) t 1 (x + s) s 1 (x x 1 ) (x x )(x x 3 ) 3 Laplacian Eigenvalues The aim of this section is to find the Laplacian spectrum of NC-graph of three groups P SL( k ) GL( q) QD n

7 M Ghorbani et al / J Linear Topological Algebra 06(0) (017) Theorem 31 The Laplacian spectrum of NC-graph of the projective special linear group P SL( k ) where k is given by {[0] 1 [u 3 u 1] t(u 1) [u 3 u] (u )(u+1) [u 3 u + 1] s(u 3) [u 3 u 1] t+s+u } where u = k s = k 1 ( k + 1) t = k 1 ( k 1) Proof By Theorem the degree of each element in the centralizer of the form xp x 1 xax 1 xbx 1 is u 3 u u 3 u + 1 u 3 u 1 respectively Let z = u 3 u p = u 1 q = u X = x z L = XI J L = (X 1)I J then det(xi ( M)) = det(a)det(d CA 1 B) where L p 0 0 L p 0 L p A = 0 L q 0 L q 0 L q (u+1)+s (J (X + 1)I) p u B = (J (X + 1)I) q u ((u+1)+s) t (u + 1)J u p J u p J u p sj u q J u q J u q C = t (u+s+1) ((X + 1)I + (t 1)J) u J u J u 0 ((X + 1) J) u 0 D = ((X + 1) J) u t

8 14 M Ghorbani et al / J Linear Topological Algebra 06(0) (017) We have det(a) = det((xi J) p ) u+1 det(((x 1)I J) q ) s = X (u+1)(u ) (X (u 1)) u+1 (x 1) s(u 3) (X 1 (u )) s = (x u 3 + u) (u )(u+1) (x u 3 + u 1) s(u 3) (x u 3 + u + 1) u+1+s thus det(d CA 1 B) = x(x u 3 + u + 1) t(u 1) (x u 3 + u + 1) t 1 = x(x u 3 + u) (u+1)(u ) (x u 3 + u 1) s(u 3) (x u 3 + u + 1) s+t+u (x u 3 + u + 1) t(u 1) Theorem 3 The Laplacian spectrum of the NC-graph of general linear group GL( q) where q = p n > p is prime integer is {[0] 1 [(q 3 q 1)(q 1)] s(s 1) [(q 3 q)(q 1)] t(k 1) [(q 3 q + 1)(q 1)] u(l 1) [(q 3 q 1)(q 1)] u+t+s 1 } where u = q(q+1) s = q(q 1) t = q + 1 l = (q 1)(q ) k = (q 1) Proof By Theorem 3 the degree of each element in centralizer of the form gdg 1 gp Z(GL( q))g 1 gig 1 is (q 1)(q 3 q +1) (q 1)(q 3 q) (q 1)(q 3 q 1) respectively Let R = (q 1)(q 3 q) N r = (R+r)I X = x R L = (X (q 1))I J L = XI J In order to find the eigenvalues of M we compute det(xi ( M)) which can easily comes to the following form det(xi ( M)) = A B C D = det(a)det(d CA 1 B)

9 M Ghorbani et al / J Linear Topological Algebra 06(0) (017) where L l 0 0 L l 0 L A = l 0 L k 0 L k 0 L k u+t (J (X + (q 1))I) l s B = (J (X + (q 1))I) k s (u+t) s uj s l J s l J s l tj s k J s k J s k C = s (u+t) ((X + (q 1))I + (s 1)J) s J s J s D = ((X + (q 1))I J) s s We have det(a) = (det(l l )) u (det(l k ))t = ((X (q 1)) l 1 (X (q 1) l)) u (X k 1 (X k)) t = (x (q 1)(q 3 q + 1)) u(l 1) (x (q 1)(q 3 q 1)) u+t (x (q 1)(q 3 q)) (k 1)t It is not difficult to see that det(d CA 1 B) = (X + R)(X + (q 1)) s(s 1) (X + (q 1) s) s 1 = x(x (q 1)(q 3 q 1)) s(s 1) (x (q 1)(q 3 q 1)) s 1 the result follows

10 144 M Ghorbani et al / J Linear Topological Algebra 06(0) (017) Theorem 33 The Laplacian spectrum of the NC-graph of quasi-dihedral group QD n is {[0] 1 [ n 1 ] n 1 3 [ n 4] n [ n ] n } Proof By Theorem 1 the NC-graph of group QD n has the following parts V 0 = {a a a n 1 a n +1 a n 1 1 } V j = {a j b a j+n b} for 1 j n Each element in part V 0 has degree n 1 every element in each V j has degree n 4 Let X = x n 1 X = x n + 4 In order to find the Laplacian spectrum of Γ(QD n) we compute det(xi ( M)) as follows: It can be easily seen that: XI 0 0 J J 0 XI 0 J J det(xi ( M)) = 0 XI J J J J X I J J J J X I det(xi ( M)) = A B C D = det(a)det(d CA 1 B) where XI XI 0 A = 0 XI n 1 J J J 0 0 B = 0 0 p p p+ ( n 1)J J J 0 0 C = 0 0 p+ p

11 M Ghorbani et al / J Linear Topological Algebra 06(0) (017) X I + ( n 1)J J J 0 X I J 0 D = 0 X I J p+ We have det(a) = (det(xi )) n 1 = ((x n 1 ) ) n 1 = (x n 1 ) n 1 det(d CA 1 B) = x(x n + ) n (x n + 4) n 1 x n 1 Therefore det(xi ( M)) = det(a)det(d CA 1 B) = x(x n 1 ) n 1 3 (x n + 4) n (x n + ) n Hence the result follows Here we find the energy of NC-graph of three groups P SL( k ) GL( q) QD n Theorem 34 The energy of NC-graph of group QD n is E(Γ(QD n)) = n 1 + (5 n 1)( n 1) Proof By using Theorem 1 we have E(Γ(QD n)) = ( n 1) + 1 n 1 k + 1 n 1 + k = n 1 n k + n 1 + k = n 1 + k = n 1 + (5 n 1)( n 1) Theorem 35 E(Γ(P SL( k ))) = 3k k+ + + α where α = x 1 + x + x 3 x 1 x x 3 are given in Theorem Theorem 36 The energy of the NC-graph of general linear group GL( q) where q = p n > p is prime integer is E(Γ(P SL( k ))) = (q 1)(q 3 4q + ) + β where β = x 1 + x + x 3 Proof The proof follows from Theorem 3

12 146 M Ghorbani et al / J Linear Topological Algebra 06(0) (017) References [1] A Abdollahi S Akbari H R Maimani Non-commuting graph of a group J Algebra 98 (006) [] M R Darafsheh Groups with the same non-commuting graph Discrete Appl Math 157 (009) [3] M Ghorbani Z Gharavi-AlKhansari An algebraic study of non-commuting graphs Filomat 31 (017) [4] I Gutman The energy of a graph Ber Math Statist Sekt Forschungsz Graz 103 (1978) 1- [5] A R Moghaddamfar W J Shi W Zhou A R Zokayi On the non-commuting graph associated with a finite group Siberian Math J 46 (005) [6] G L Morgan C W Parker The diameter of the commuting graph of a finite group with trivial centre J Algebra 393 (013) [7] B H Neumann A problem of Paul Erdős on groups J Austral Math Soc Ser A 1 (1976)

A Note on Integral Non-Commuting Graphs

Filomat 31:3 (2017), 663 669 DOI 10.2298/FIL1703663G Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A Note on Integral Non-Commuting