SIGNLESS AND NORMALIZED LAPLACIAN SPECTRUMS OF THE POWER GRAPH AND ITS SUPERGRAPHS OF CERTAIN FINITE GROUPS

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1 J Idoes Math Soc Vol No (08) pp 6 69 SIGNLESS AND NORMALIZED LAPLACIAN SPECTRUMS OF THE POWER GRAPH AND ITS SUPERGRAPHS OF CERTAIN FINITE GROUPS A Hamzeh Departmet of Pure Mathematics Faculty of Mathematical Scieces Uiversity of Kasha Kasha I R Ira Abstract The aim of this article is to compute the sigless ad ormalized Laplacia spectrums of the power graph its mai supergraph ad cyclic graph of dihedral ad dicyclic groups Key words ad Phrases: Power graph sigless Laplacia ormalized Laplacia cyclic graph mai supergraph Abstrak Tujua dari paper ii adalah utuk meghitug spektrum Laplasia tapa tada da diormalka dari graf pagkat super graf utamaya da graf siklis dari grup disiklis da dihedral Kata kuci: Graf pagkat Laplasia tapa tada Laplasia diormalka graf siklis supergraf utama Basic Cocepts All groups i this paper are assumed to be fiite ad graphs are simple Let Γ be a graph with vertex set V (Γ) ad edge set E(Γ) For the vertex i i V (Γ) d Γ (i) is degree of i that is equal to the umber of eighbors of i If each vertex of graph has the same degree of r the graph is r-regular The adjacecy matrix A(Γ) Laplacia matrix L(Γ) sigless Laplacia matrix Q(Γ) ad ormalized Laplacia 000 Mathematics Subject Classificatio: 05C5 05C50 Received: revised: accepted:

2 6 A Hamzeh matrix L(Γ) are defied as follows: ij E(Γ) A(Γ) = (a ij ) = 0 otherwise L(Γ) = D(Γ) A(Γ) Q(Γ) = D(Γ) + A(Γ) i = j ad d Γ (i) 0 L(Γ) = (L ij ) = ij E(Γ) dγ(i)d Γ(j) 0 otherwise where D(Γ) is the diagoal matrix of degrees If N is a square matrix the P (N x) = det(xin) is characteristic polyomial of N The multi-sets of all eigevalues Laplacia eigevalues sigless Laplacia eigevalues ad ormalized Laplacia eigevalues of Γ are deoted by Spec(Γ) Spec L (Γ) Spec Q (Γ) ad Spec L (Γ) respectively We usually write Spec(Γ) = {λ (s) λ (sm) m } where λ λ m are distict Γ-eigevalues ad s j is the multiplicity of λ j j m Followig Sabidussi [7 p 96] the A-joi of a set of graphs {Γ a } a A is the graph with the followig vertex ad edge sets: V ( ) = {(x y) x V (A) & y V (Γ x )} E( ) = {(x y)(x y ) xx E(A) or else x = x & yy E(Γ x )} For the p-vertex graph A the Ajoi of H H H p is deoted by A[H H H p ] Let G be a fiite group For x G the order of x is deoted by o(x) The set of all elemet orders of G is deoted by π e (G) ad the umber of all elemets of G of order i is deoted by Ω i (G) The ivestigatio of graphs related to algebraic structures is importat because graphs like these have importat applicatios (see for example [0]) ad are related to automata theory (see [8]) The Cayley graph is the oldest simple graphs associated to the fiite group G The power graph was itroduced by Kelarev ad Qui i [] Two elemets x y G are adjacet i the power graph if ad oly if oe is a power of the other Camero ad Ghosh i [] proved that abelia groups with the same umber of elemets of each possible order ca be characterized by their power graphs For more iformatio o power graphs see [ 9 ] The cyclic graph Γ G is a simple graph with the vertex set G Two elemets x y G are adjacet i the cyclic graph if ad oly if x y is cyclic [] Set π e (G) = {a a k } ad defie the graph G with vertex set π e (G) ad edge set E( G ) = {xy x y π e (G) x y or y x} I [ 5] the preset author itroduced the mai supergraph S(G) that is a graph with vertex set G i which two vertices x ad y are adjacet if ad oly if o(x) o(y) or o(y) o(x) I the metioed papers they proved that S(G) = G [K Ωa (G) K Ωak (G)] where K deotes the complete graph o vertices Note that the graphs S(G) ad Γ G are supergraphs of the power graph Laplacia spectrum of power graphs of

3 Sigless ad Normalized Laplacia Spectrums 6 fiite cyclic ad dihedral groups computed by Chattopadhyaya ad Paigrahi i [] ad i [5] Mehraia et al obtaied spectrum of the power graphs of D T ad SD 8 Also i [6 7] the eigevalues ad Laplacia eigevalues of S(G) are computed Throughout this paper we refer to [6] for group theory cocepts ad for graph theoretical cocepts ad otatios we refer to [9] I this paper we compute sigless Laplacia ad ormalized Laplacia eigevalues of the power graph ad some of its supergraphs As a applicatio sigless Laplacia ad ormalized Laplacia eigevalues of the power graph ad its supergraphs are computed for the dihedral ad dicyclic groups Mai Results The dihedral ad dicyclic groups ca be preseted as follows: D = < a b a = b = bab = a > T = < a b a = a = b b ab = a > We ow state a result of [0] which is importat i our ext results Theorem [0] Let H be a graph with V (H) = { k} ad G i s be r i - regular graphs of order i (i = k) If Γ = H[G G G k ] the sigless Laplacia spectrum ca be computed as follows: where ad ( k Spec Q (Γ) = (N i + (Spec Q (G i )\{r i }))) Spec((CQ (H)) i= j N H (i) j N H (i) N i = 0 otherwise r i + N i i = j C Q (H) = (c ij ) k k = i j ij E(H) 0 otherwise The followig result is a immediate cosequece of Theorem ad the fact that Spec Q (K ) = { () } Corollary Suppose S(G) = G [K Ωa (G) K Ωak (G)] The the sigless Laplacia spectrum of the mai supergraph is computed as follows: ( k ) Spec Q (S(G)) = (N i + Ω ai (G) ) (Ωa i (G)) Spec(C) i=

4 6 A Hamzeh where a j N G (a i) Ω a j (G) N G (a i ) N i = 0 otherwise Ωal (G)Ω as (G) a l a s or a s a l ρ ls = ρ sl = 0 otherwise ad Ω a (G) + N ρ ρ k ρ Ω a (G) + N ρ k C = ρkk ρ k ρ k Ω ak (G) + N k I the ext results the sigless Laplacia spectrum of the power graph of some fiite groups are computed Cosider the dihedral group D I this case Γ D = P [K K K ] If is a prime power the P(D ) = P [K K K ] Corollary The sigless Laplacia eigevalues of Γ D ad P(D ) where i this case is a prime power are with multiplicity with multiplicity ad three simple sigless Laplacia eigevalues as follows: x = ( 9 + ) x = 6 ( ) ± i ( ( 9 + ) )

5 Sigless ad Normalized Laplacia Spectrums 65 Proof By Theorem C Q (P ) = 0 0 Now by computig eigevalues of the matrix C Q (P ) the result will be completed Corollary The sigless Laplacia eigevalues of Γ T ad P(T ) where i this case is a power of are with multiplicity with multiplicity with multiplicity with multiplicity ad three simple sigless Laplacia eigevalues as follows: x = ( ) x = + ( ) ± i ( ( ) ) Proof A simple ivestigatio of dicyclic group T shows that Γ T = W [K K K K K }{{} ] where W is depicted i Figure If is a power of the P(T ) = W [K K K K K }{{} ]

6 66 A Hamzeh Figure The graph W related to the cyclic graph ad power graph of T So by Theorem C Q (W ) = Now by computig eigevalues of the matrix C Q (W ) the proof is completed Theorem 5 (See Tamizh Chelvam ad Sattaatha [8]) Let G be a elemetary abelia group of order p for some prime umber p ad positive iteger The P(G) = K + l i= K p where l = p p Let E(p ) deote the elemetary abelia group of order p p is prime is positive iteger ad l = p p By Theorem 5 P(E(p )) = K + l i= K p ad so P(E(p )) = M[K K p K p ] }{{} l where M = K + K l Corollary 6 The sigless Laplacia eigevalues of E(p ) are p with multiplicity l(p ) p with multiplicity l ad two simple sigless Laplacia eigevalues as follows: x = + lp l 9 + lp 0l p + + p ± l p l p lp + l + p

7 Sigless ad Normalized Laplacia Spectrums 67 Proof By Theorem ad the structure of P(E(p )) l(p ) p p p p p p p p 0 p C Q (M) = p 0 0 p 0 0 p p 0 p p Now by computig eigevalues of the matrix C Q (M) the result is obtaied We ow state aother result of [0] which is crucial i computig ormalized Laplacia eigevalues Γ D P(D ) Γ T ad P(T ) Theorem 7 [0] Let H be a graph with o isolated vertices ad V (H) = { k} ad G i s be r i -regular graphs of order i (i = k) If Γ = H[G G G k ] the ormalized Laplacia spectrum ca be computed as follows: where ad ( k N i Spec L (Γ) = ( + r i + N i i= r i r i + N i (Spec L (G i )\{0})) j N H (i) j N H (i) N i = 0 otherwise C L (H) = (c ij ) k k = N i r i+n i i = j i j (r i+n i)(r j+n j) ) Spec((CL (H)) ij E(H) 0 otherwise The followig results are immediate cosequeces of Theorem 7 ad the fact that Spec L (K ) = {0 () } Corollary 8 The ormalized Laplacia eigevalues of Γ D ad P(D ) where i this case is a prime power are with multiplicity with multiplicity 0 with multiplicity ad two simple ormalized Laplacia eigevalues as follows: x = + + ± ( + )

8 68 A Hamzeh Proof Cosider the dihedral group D I this case Γ D = P [K K K ] If is a prime power the P(D ) = P [K K K ] By Theorem 7 0 C L (P ) = 0 Now by computig eigevalues of the matrix C L (P ) we will prove the result Corollary 9 The ormalized Laplacia eigevalues of Γ T ad P(T ) where i this case is a power of are with multiplicity with multiplicity with multiplicity ad Spec(C) that C = () ()() () ()() () () () () () () () () Proof the proof is similar to Corollary 8 p p Corollary 0 The ormalized Laplacia eigevalues of E(p ) are with multiplicity l(p ) + p with multiplicity l ad 0 with multiplicity Proof By Theorem 7 ad the structure of P(E(p )) l(p) l(p) l(p) l(p) l(p) p l(p) 0 p C L (M) = l(p) 0 0 p 0 0 l(p) p 0 l(p) p Now by computig eigevalues of this matrix the proof is completed l(p) Ackowledgemet This research is supported by Natioal Elites Foudatio uder Grat Number 5/80

9 Sigless ad Normalized Laplacia Spectrums 69 Refereces [] Camero PJ ad Ghosh S The power graph of a fiite group Discrete Math (0) 0 [] Camero PJ The power graph of a fiite group II J Group Theory (00) [] Chattopadhyaya S ad Paigrahi P O Laplacia spectrum of power graphs of fiite cyclic ad dihedral groups Liear Multiliear Algebra 6:7(05) 5 55 [] Hamzeh A ad Ashrafi AR The mai supergraph of the power graph of a fiite group submitted [5] Hamzeh A ad Ashrafi AR Automorphism group of supergraphs of the power graph of a fiite group Europea J Combi 60(07) 8 88 [6] Hamzeh A Spectrum ad L-spectrum of the cyclic group Southeast Asia Bull Math accepted [7] Hamzeh A ad Ashrafi AR Spectrum ad Lspectrum of the power graph ad its mai supergraph for certai fiite groups submitted [8] Kelarev A Rya J ad Yearwood J Cayley graphs as classifiers for data miig: The i uece of asymmetries Discrete Math 09:7(009) [9] Kelarev AV ad Qui SJ A combiatorial property ad power graphs of semigroups Commet Math Uiv Caroli 5:(00) 7 [0] Kelarev AV Graph Algebras ad Automata Marcel Dekker New York 00 [] Kelarev AV ad Qui SJ Directed graphs ad combiatorial properties of semigroups J Algebra 5:(00) 6 6 [] Kelarev AV Qui SJ ad Smolíková R Power graphs ad semigroups of matrices Bull Austral Math Soc 6:(00) [] Kelarev AV ad Qui SJ A combiatorial property ad power graphs of groups Cotributios to Geeral Algebra (Viea 999) 9 5 Hey Klagefurt 000 [] Ma XL Wei HQ ad Guo Zhog The cyclic graph of a fiite group Algebra 0(0) Article ID [5] Mehraia Z Gholami A ad Ashrafi AR The Spectra of power graphs of certai fiite groups Liear Multiliear Algebra 65:5(07) [6] Rose JS A Course o Group Theory Cambridge Uiversity Prees Cambridge New York- Melboure 978 [7] Sabidussi G Graph Derivatives Math Z 76(96) 85 0 [8] Tamizh Chelvam T ad Sattaatha M Power graph of fiite abelia groups Algebra Discrete Math 6:(0) [9] West DB Itroductio to Graph Theory Secod Editio Pretice Hall Ic Upper Saddle River NJ 00 [0] Wu B-F Lou Y-Y ad He C-X Sigless Laplacia ad ormalized Laplacia o the H-joi operatio of graphs Discrete Math Algorithm Appl 06(0) [ pages] DOI: