On Laplacian energy of non-commuting graphs of finite groups

Transcription

1 Journal of Linear and Topological Algebra Vol 07, No 0, 018, On Laplacian energy of non-commuting graphs of finite groups P Dutta a, R K Nath a, a Department of Mathematical Sciences, Tezpur University, Napaam-78408, Sonitpur, Assam, India Received 13 December 017; Accepted 30 April 018 Communicated by Hamidreza Rahimi Abstract Let be a finite non-abelian group with center Z( The non-commuting graph of is a simple undirected graph whose vertex set is \ Z( and two vertices x and y are adjacent if and only if xy yx In this paper, we compute Laplacian energy of the noncommuting graphs of some classes of finite non-abelian groups c 017 IAUCTB All rights reserved Keywords: Non-commuting graph, L-spectrum, Laplacian energy, finite group 010 AMS Subject Classification: 0D60, 05C50, 15A18, 05C5 1 Introduction Let be a graph Let A( and D( denote the adjacency matrix and degree matrix of respectively Then the Laplacian matrix of is given by L( = D( A( Let β 1, β,, β m be the eigenvalues of L( with multiplicities b 1, b,, b m Then the Laplacian spectrum of, denoted by L-spec(, is the set {β b 1 1, βb,, βbm m } The Laplacian energy of, denoted by LE(, is given by LE( = µ L-spec( µ e( v( (1 where v( and e( are the sets of vertices and edges of respectively It is worth mentioning that the notion of Laplacian energy of a graph was introduced by utman Corresponding author address: parama@gonitsoracom (P Dutta; rajatkantinath@yahoocom (R K Nath Print ISSN: c 017 IAUCTB All rights reserved Online ISSN:

2 1 P Dutta and R K Nath / J Linear Topological Algebra 07(0 ( and Zhou [19] A graph is called L-integral if L-spec( contains only integers Various properties of L-integral graphs and LE( are studied in [, 1, 3, 31, 3] Let be a finite non-abelian group with center Z( The non-commuting graph of, denoted by A, is a simple undirected graph such that v(a = \Z( and two vertices x and y are adjacent if and only if xy yx Various aspects of non-commuting graphs of different families of finite non-abelian groups are studied in [1, 3, 9, 17, 30] Note that the complement of A is the commuting graph of denoted by A Commuting graphs of finite groups are studied extensively in [4, 1 14, 0, 4, 7, 8] In [11], Dutta et al have computed the Laplacian spectrum of the non-commuting graphs of several well-known families of finite non-abelian groups In this paper we compute the Laplacian energy of those classes of finite groups It is worth mentioning that horbani and haravi- Alkhansari [18] have computed the energy of non-commuting graphs of the projective special linear group P SL(, k, where k, the general linear group L(, q, where q = p n (p is a prime and n 4 and the quasi-dihedral group QD n recently roups with known central factors In this section, we compute Laplacian energy of some families of finite groups whose central factors are well-known Theorem 1 Let be a finite group and Z( = Sz(, where Sz( is the Suzuki group presented by a, b : a 5 = b 4 = 1, b 1 ab = a Then LE(A = Proof It is clear that = 19 Z( Since ( 10 Z( + 30 Z( 19 Z( = Sz(, we have Z( = az(, bz( : a5 Z( = b 4 Z( = Z(, b 1 abz( = a Z( Note that for any z Z(, C (a = C (az = Z( az( a Z( a 3 Z( a 4 Z(, C (ab = C (abz = Z( abz( a 4 b Z( a 3 b 3 Z(, C (a b = C (a bz = Z( a bz( a 3 b Z( ab 3 Z(, C (a b 3 = C (a b 3 z = Z( a b 3 Z( ab Z( a 4 bz(, C (b = C (bz = Z( bz( b Z( b 3 Z( and C (a 3 b = C (a 3 bz = Z( a 3 bz( a b Z( a 4 b 3 Z( are the only centralizers of non-central elements of Since all these distinct centralizers are abelian, we have A = K 4 Z( 5K 3 Z( and hence e(a = 150 Z( By Theorem 31 of [11], we have L-spec(A = {0, (15 Z( 4 Z( 1, (16 Z( 15 Z( 5, (19 Z( 5 }

3 P Dutta and R K Nath / J Linear Topological Algebra 07(0 ( So, 0 e(a = Z( 19 Z(, e(a = Z( 19 Z(, e(a = 4 19 Z( 19 Z( and e(a = Z( By (1, we have Hence, the result follows LE(A = 300 ( 15 Z( + (4 Z( 19 ( 4 + (15 Z( 5 19 Z( Theorem Let be a finite group such that Then 19 Z( + 5 LE(A = p(p Z( ( Z( Z( = Z p Z p, where p is a prime Z( = Proof It is clear that = (p Z( Since Z( = Z p Z p, we have az(, bz( : a p, b p, aba 1 b 1 Z(, where a, b with ab ba Then for any z Z(, C (a = C (a i z = Z( az( a p 1 Z( for 1 i p and C (a j b = C (a j bz = Z( a j bz( a j b p 1 Z( for 1 j p are the only centralizers of non-central elements of Also note that these centralizers are abelian subgroups of Therefore A = K C(a\Z( ( p j=1 K C(a\Z( Since, C (a = C (a j b = p Z( for 1 j p, we have A = (p + 1K (p 1 Z( and hence e(a = p(p+1(p 1 Z( By Theorem 3 of [11], we have L-spec(A = {0, ((p p Z( (p 1 Z( p 1, ((p Z( p } Therefore, 0 e(a = p(p Z(, (p p Z( e(a = 0 and (p Z( e(a = (p Z( By (1, we have LE(A = p(p Z( + ((p Z( p 0 + p((p Z( Hence the result follows Corollary 3 Let be a non-abelian group of order p 3, for any prime p Then LE(A = p (p Proof Note that Z( = p and Z( = Z p Z p Hence the result follows from Theorem

4 14 P Dutta and R K Nath / J Linear Topological Algebra 07(0 ( Theorem 4 Let be a finite group such that Z( = D m, for m Then LE(A = (m 3m(m Z( + m(4m 3 Z( m Proof Clearly, = (m Z( Since Z( = D m we have Z( = xz(, yz( : x, y m, xyx 1 y Z(, where x, y with xy yx It is easy to see that for any z Z(, C (xy j = C (xy j z = Z( xy j Z(, 1 j m and C (y = C (y i z = Z( yz( y m 1 Z(, 1 i m are the only centralizers of non-central elements of Also note that these centralizers are abelian subgroups of and C (xy j = Z( for 1 j m and C (y = m Z( Hence A = K (m 1 Z( mk Z( and e(a = 3m(m 1 Z( By Theorem 34 of [11], we have L-spec(A = {0, (m Z( (m 1 Z( 1, ((m Z( m Z( m, ((m Z( m } Therefore, 0 e(a 3m(m Z( =, m m Z( e(a m(m Z( =, m (m Z( e(a (m (m Z( =, m (m Z( e(a = (m m + 1 Z( m By (1, we have ( 3m(m Z( m(m Z( LE(A = + ((m Z( m m ( ( (m (m Z( (m m + 1 Z( + (m Z( m + m m m and hence, the result follows Using Theorem 4, we now compute the Laplacian energy of the non-commuting graphs of the groups M mn, D m and Q 4n respectively

5 P Dutta and R K Nath / J Linear Topological Algebra 07(0 ( Corollary 5 Let M mn = a, b : a m = b n = 1, bab 1 = a 1 be a metacyclic group, where m > Then LE(A Mmn = { m(m 3(m 1n +m(4m 3n m 1, if m is odd m(m (m 3n +m(m 3n m 1, if m is even Proof Observe that Z(M mn = b or b a m b according as m is odd or even M Also, it is easy to see that mn Z(M mn = D m or D m according as m is odd or even Hence, the result follows from Theorem 4 Corollary 6 Let D m = a, b : a m = b = 1, bab 1 = a 1 be the dihedral group of order m, where m > Then { m, if m is odd LE(A Dm = m(m 3m+3 m 1, if m is even Corollary 7 Let Q 4m = x, y : y m = 1, x = y m, yxy 1 = y 1, where m, be the generalized quaternion group of order 4m Then LE(A Q4m = m(4m 6m + 3 m Proof The result follows from Theorem 4 noting that Z(Q 4m = {1, a m } and Q 4m Z(Q 4m = D m 3 Some well-known groups In this section, we compute Laplacian energy of the non-commuting graphs of some well-known families of finite non-abelian groups Proposition 31 Let be a non-abelian group of order pq, where p and q are primes with p (q Then LE(A = q(p (q pq Proof It is clear that = pq Note that Z( = 1 and the centralizers of non-central elements of are precisely the Sylow subgroups of The number of Sylow q-subgroups and Sylow p-subgroups of are one and q respectively Therefore, we have A = K q 1 qk p 1 and hence e(a = q(p 1(q 1 By Proposition 41 of [11], we have L-spec(A = {0, (pq q q, (pq p pq q, (pq q }

6 16 P Dutta and R K Nath / J Linear Topological Algebra 07(0 ( So, 0 e(a = p q p q q +q pq 1, pq q e(a = q(q p(p 1 pq 1, pq p e(a = and pq e(a = p q+q pq q+1 By (1, we have (q p(q 1 pq 1 pq 1 LE(A = p q p q q ( + q q(q p(p + (q pq pq ( ( (q p(q p q + q pq q (pq q + q pq pq and hence, the result follows Proposition 3 Let QD n denotes the quasidihedral group a, b : a n 1 = b = 1, bab 1 = a n 1, where n 4 Then LE(A QD n = 3n 3 n + 3 n n 1 Proof It is clear that Z(QD n = {1, a n }; and so v(a QD n = ( n 1 Note that C QD n (a = C QD n (a i = a for 1 i n 1, i n C QD n (a j b = {1, a n, a j b, a j+n b} for 1 j n and are the only centralizers of non-central elements of QD n Note that these centralizers are abelian subgroups of QD n Therefore, we have A QD n = K CQD n (a\z(qd n ( n K C j=1 QD n (a j b\z(qd n Since C QD n (a = n 1 and C QD n (a j b = 4 for 1 j n, we have A QD n = K n 1 n K Hence By Proposition 4 of [11], we have e(a QD n = 3n 6 n 1 L-spec(A QD n = {0, ( n 1 n 1 3, ( n 4 n, ( n n } Therefore, 0 e(a QD n v(a QD n = 3n 1 ( n 1 n 1, n 1 e(a QD n v(a QD n = n 4 n 1 n 1, n 4 e(aqd n v(a QD n = n 6 n 1 +8 n 1 and n e(aqd n v(a QD n = n n 1 +4 n 1 By (1, we have LE(A QD n = 3n 1 ( n 1 n 1 + n ( n 6 n n 1 ( + ( n 1 n 4 n 1 3 n 1 + n ( n n n 1

7 P Dutta and R K Nath / J Linear Topological Algebra 07(0 ( and hence, the result follows Proposition 33 Let denotes the projective special linear group P SL(, k, where k Then LE(A = 36k 5k 7 4k + 3k + 4 k + k 3k k Proof We have = 3k k 1, since is a non-abelian group of order k ( k 1 with trivial center By Proposition 31 of [1], the set of centralizers of non-trivial elements of is given by {xp x 1, xax 1, xbx 1 : x } where P is an elementary abelian -subgroup and A, B are cyclic subgroups of having order k, k and k + 1 respectively Also the number of conjugates of P, A and B in are k + 1, k 1 ( k + 1 and k 1 ( k respectively Hence A is given by ( k + 1K xp x 1 1 k 1 ( k + 1K xax 1 1 k 1 ( k K xbx 1 1 That is, A = ( k + 1K k 1 k 1 ( k + 1K k k 1 ( k K k Therefore, By Proposition 43 of [11], we have e(a = 6k 3 4k 3k + k + k L-spec(A = {0,( 3k k+1 3k 1 k + k 1, ( 3k k+1 k k, ( 3k k k 1 k 3 k 1, ( 3k k k + k } Now, 0 e(a = 6k 3 4k 3k + k + k 3k k 1, 3k k+1 e(a = 3k k 1 3k k+1 e(a = k 3k k 1, 3k k e(a = 3k 1 3k k 1 and 3k k e(a = 4k 3k k + k +1 3k k 1 By (1, we have v(a 3k k 1, LE(A = 6k 3 4k 3k + k + k ( 3k k + ( 3k 1 k + k 1 3k k 3k k ( + ( k k k ( 3k k + ( 3k 1 k 3 k 1 3k 3k k ( + ( k + k 4k 3k k + k + 1 3k k and hence, the result follows

8 18 P Dutta and R K Nath / J Linear Topological Algebra 07(0 ( Proposition 34 Let denotes the general linear group L(, q, where q = p n > and p is a prime Then LE(A = q9 q 8 q 7 + q 6 + q 5 + q 4 4q 3 + q + q q 4 q 3 q + 1 Proof We have = (q (q q and Z( = q Therefore, = q 4 q 3 q + 1 By Proposition 36 of [1], the set of centralizers of non-central elements of L(, q is given by {xdx 1, xix 1, xp Z(L(, qx 1 : x L(, q} where D is the subgroup of L(, q consisting of all diagonal matrices, I is a cyclic subgroup of L(, q having order q and P is the Sylow p-subgroup of L(, q consisting of all upper triangular matrices with 1 in the diagonal The orders of D and P Z(L(, q are (q and q(q respectively Also the number of conjugates of D, I and P Z(L(, q in L(, q are q(q+1, q(q 1 and q + 1 respectively Hence the commuting graph of L(, q is given by q(q + 1 K xdx 1 q+1 q(q K xix 1 q+1 (q + 1K xp Z(L(,qx 1 q+1 Thus, A = q(q+1 K q 3q+ q(q 1 K q q (q + 1K q q+1 Hence, e(a = q 8 q 7 q 6 +5q 5 +q 4 4q 3 +q By Proposition 44 of [11], we have L-spec(A = {0,(q 4 q 3 q + q q3 q q, (q 4 q 3 q + q + 1 q 4 q 3 +q, (q 4 q 3 q + 3q q 4 q 3 q +q, (q 4 q 3 q + 1 q +q } So, 0 e(a v(a = q8 q 7 q 6 +5q 5 +q 4 4q 3 +q q 4 q 3 q +1, q 4 q 3 q + q e(a v(a = q3 q +q q 4 q 3 q +1, q 4 q 3 q + q + 1 e(a v(a = q5 q 4 q 3 +3q 1 q 4 q 3 q +1, q 4 q 3 q + 3q e(a v(a = q 5 q 4 +q 3 q +q 1 q 4 q 3 q +1 and q 4 q 3 q + 1 e(a = q6 3q 5 +q 4 +q 3 q q+1 q 4 q 3 q +1 By (1, we have LE(A = q8 q 7 q 6 + 5q 5 + q 4 4q 3 ( + q q q 4 q 3 q + (q 3 q 3 q + q q + 1 q 4 q 3 q + 1 ( q 4 q 3 ( + q q 5 q 4 q 3 + 3q + q 4 q 3 q + 1 ( q 4 q 3 q ( + q q 5 q 4 + q 3 q + q + q 4 q 3 q + 1 and hence, the result follows

9 P Dutta and R K Nath / J Linear Topological Algebra 07(0 ( Proposition 35 Let F = F ( n, n and ϑ be the Frobenius automorphism of F, ie, ϑ(x = x for all x F If denotes the group U(a, b = a 1 0 : a, b F b ϑ(a 1 under matrix multiplication given by U(a, bu(a, b = U(a + a, b + b + a ϑ(a, then LE(A = n+1 n+ Proof Note that Z( = {U(0, b : b F } and so Z( = n Therefore, = n ( n Let U(a, b be a non-central element of The centralizer of U(a, b in is Z( U(a, 0Z( Hence A = ( n K n and e(a = 4n 3 3n + n By Proposition 45 of [11], we have L-spec(A = {0, ( n n+1 (n 1, ( n n n } Thus, 0 e(a = n n, n n+1 e(a = 0 and n n e(a = n By (1, we have LE(A = n n + (( n 0 + ( n n and hence, the result follows Proposition 36 Let F = F (p n where p is a prime If denotes the group V (a, b, c = a 1 0 : a, b, c F b c 1 under matrix multiplication V (a, b, cv (a, b, c = V (a + a, b + b + ca, c + c, then LE(A = (p 3n p n Proof We have Z( = {V (0, b, 0 : b F } and so Z( = p n Therefore, = p n (p n The centralizers of non-central elements of A(n, p are given by (1 If b, c F and c 0 then the centralizer of V (0, b, c in is {V (0, b, c : b, c F } having order p n ( If a, b F and a 0 then the centralizer of V (a, b, 0 in is {V (a, b, 0 : a, b F } having order p n (3 If a, b, c F and a 0, c 0 then the centralizer of V (a, b, c in is {V (a, b, ca a 1 : a, b F } having order p n It can be seen that all the centralizers of non-central elements of A(n, p are abelian Hence, A = K p n p K n p n p n (pn K p n p = n (pn + 1K p n p n

10 130 P Dutta and R K Nath / J Linear Topological Algebra 07(0 ( and e(a = p6n p 5n p 4n +p 3n By Proposition 46 of [11], we have L-spec(A A(n,p = {0, (p 3n p n p3n p n 1, (p 3n p n pn } So, 0 e(a = p 3n p n, p n p n By (1, we have p 3n p n e(a = 0 and LE(A = p 3n p n + (p 3n p n 0 + p n (p n p n p 3n p n e(a = and hence, the result follows 4 Some consequences In this section, we derive some consequences of the results obtained in Section and Section 3 For a finite group, the set C (x = {y : xy = yx} is called the centralizer of an element x Let Cent( = {C (x : x }, that is the number of distinct centralizers in A group is called an n-centralizer group if Cent( = n The study of these groups was initiated by Belcastro and Sherman [6] in the year 1994 The readers may conf [10] for various results on these groups We begin with computing Laplacian energy of non-commuting graphs of finite n-centralizer groups for some positive integer n It may be mentioned here that various energies of commuting graphs of finite n-centralizer groups have been computed in [15] Proposition 41 If is a finite 4-centralizer group, then LE(A = 4 Z( Proof Let be a finite 4-centralizer group Then, by Theorem of [6], we have Z Z Therefore, by Theorem, the result follows Further, we have the following result Corollary 4 If is a finite (p + -centralizer p-group for any prime p, then Z( = LE(A = p(p Z( Proof Let be a finite (p + -centralizer p-group Then, by Lemma 7 of [5], we have Z( = Z p Z p Therefore, by Theorem, the result follows Proposition 43 If is a finite 5-centralizer group, then LE(A = 1 Z( or 18 Z( +7 Z( 5 Proof Let be a finite 5-centralizer group Then by Theorem 4 of [6], we have Z( = Z 3 Z 3 or D 6 Now, if Z( = Z 3 Z 3, then by Theorem, we have LE(A = 1 Z( If Z( = D 6, then by Theorem 4 we have LE(A = 18 Z( +7 Z( 5 This completes the proof Let be a finite group The commutativity degree of is given by the ratio Pr( = {(x, y : xy = yx}

11 P Dutta and R K Nath / J Linear Topological Algebra 07(0 ( The origin of commutativity degree of a finite group lies in a paper of Erdös and Turán (see [16] Readers may conf [7, 8, 5] for various results on Pr( In the following few results we shall compute Laplacian energy of non-commuting graphs of finite non-abelian groups such that Pr( = r for some rational number r Proposition 44 Let be a finite group and p the smallest prime divisor of If Pr( = p +p 1 p, then LE(A 3 = p(p Z( Proof If Pr( = p +p 1 p, then by Theorem 3 of [], we have 3 Z p Z p Hence the result follows from Theorem As a corollary we have the following result Z( is isomorphic to Corollary 45 Let be a finite group such that Pr( = 5 8 Then LE(A = 4 Z( Proposition 46 If Pr( { 5 14, 5, 11 7, 1 }, then LE(A = 9, , 5 or 5 Proof If Pr( { 5 14, 5, 11 7, 1 }, then as shown in [9, pp 46] and [6, pp 451], we have Z( is isomorphic to one of the groups in {D 6, D 8, D 10 D 14 } Hence the result follows from Corollary 6 Proposition 47 Let be a group isomorphic to any of the following groups (1 Z D 8 ( Z Q 8 (3 M 16 = a, b : a 8 = b = 1, bab = a 5 (4 Z 4 Z 4 = a, b : a 4 = b 4 = 1, bab 1 = a 1 (5 D 8 Z 4 = a, b, c : a 4 = b = c = 1, ab = ba, ac = ca, bc = a cb (6 S(16, 3 = a, b : a 4 = b 4 = 1, ab = b 1 a 1, ab 1 = ba 1 Then LE(A = 16 Proof If is isomorphic to any of the above listed groups, then = 16 and Z( = 4 Therefore, Z( = Z Z Thus the result follows from Theorem Recall that genus of a graph is the smallest non-negative integer n such that the graph can be embedded on the surface obtained by attaching n handles to a sphere A graph is said to be planar if the genus of the graph is zero We conclude this paper with the following result Theorem 48 If the non-commuting graph of a finite non-abelian group is planar, then LE(A = 8 3 or 9 Proof By Theorem 31 of [3], we have = D 6, D 8 or Q 8 If = D 8 or Q 8 then by Corollary 6 and Corollary 7 it follows that LE(A = 8 3 If = D 6 then, by Corollary 6, LE(A = 9 This completes the proof Acknowledgements The authors would like to thank the referee for his/her valuable comments

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