Malaya J. Mat. 4(3)(2016) Reciprocal Graphs

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1 Malaya J Mat 43)06) Reciprocal Graphs G Idulal a, ad AVijayakumar b a Departmet of Mathematics, StAloysius College, Edathua, Alappuzha , Idia b Departmet of Mathematics, Cochi Uiversity of Sciece ad Techology, Cochi-68 0, Idia Abstract Eigevalue of a graph is the eigevalue of its adjacecy matrix A graph G is reciprocal if the reciprocal of each of its eigevalue is also a eigevalue of G The Wieer idex WG) of a graph G is defied by WG) = d where D is the distace matrix of G I this paper some ew classes of reciprocal graphs d D ad a upperboud for their eergy are discussed Pairs of equieergetic reciprocal graphs o every 0 mod ) ad 0 mod 6) are costructed The Wieer idices of some classes of reciprocal graphs are also obtaied Keywords: Eigevalue, Eergy, Reciprocal graphs, splittig graph, Wieer idex 00 MSC: 39B55, 39B5, 39B8 c 0 MJM All rights reserved Itroductio Let G be a graph of order ad size m with the vertex set VG) labelled as {v, v,, v } The set of eigevalues {,,, } of a adjacecy matrix A of G is called its spectrum ad is deoted by specg) No-isomorphic graphs with the same spectrum are called cospectral Studies o graphs with a specific patter i their spectrum have bee of iterest Gutma ad Cvetkovic studied the spectral structure of graphs havig a maximal eigevalue ot greater tha i [5] ad Baliska etal have studied graphs with itegral spectra i [] I [] some ew costructios of itegral graphs are provided Dias i [6] has idetified graphs with complemetary pairs of eigevalues eigevalues ad with + = ) A graph G is reciprocal [0] if the reciprocal of each of its eigevalue is also a eigevalue of G The first referece of a reciprocal graph appeared i the work of JR Dias i [6, 7] ad the chemical molecules of Dedralee ad Radialee have bee discussed there i I [0] some classes of reciprocal graphs have bee idetified I [3] reciprocal graphs are also referred to as graphs with property R The eergy of a graph G [], deoted by EG) is the sum of the absolute values of its eigevalues No-cospectral graphs with the same eergy are called equieergetic I [8, 9, 5] some bouds o eergy are described I [] ad [, 3] a pair of equieergetic graphs are costructed for every 0 mod 4) ad 0 mod 5) ad i [0] we have exteded it for = 6, 4, 8 ad 0 I [7] a pair of equieergetic graphs withi the family of iterated lie graphs of regular graphs ad i [] a pair of equieergetic graphs obtaied from the cross product of graphs are described I [] a pair of equieergetic self-complemetary graphs o vertices is costructed for every = 4k ad = 4t +, k, t 3 A plethora of papers have bee appeared dealig with this parameter i recet years The distace matrix of a coected graph G, deoted by DG) is defied as DG) = [ dv i, v j ) ] where dv i, v j ) is the distace betwee v i ad v j The Wieer idex WG) is defied by Correspodig author address: idulalgopal@gmailcom G Idulal) ad vambat@gmailcom AVijayakumar)

2 G Idulal et al / Reciprocal Graphs 38 WG) = d The chemical applicatios of this idex are well established i [6, 8] d D I this paper, we costruct some ew classes of reciprocal graphs ad a upperboud for their eergy is obtaied Pairs of equieergetic reciprocal graphs o 0 mod ) ad 0 mod 6) are costructed The Wieer idices of some classes of reciprocal graphs are also obtaied These results are ot foud so far i literature Some ew classes of reciprocal graphs If A ad B are two matrices the A B deote the tesor product of A ad B We use the followig properties of block matrices[4] [ ] M N Lemma Let M, N, P ad Q be matrices with M ivertible Let S = The S = M Q PM N P Q Moreover if M ad P commutes the S = MQ PN where the symbol deotes the determiat We cosider the followig operatios o G Operatio Attach a pedat vertex to each vertex of G The resultat graph is called the pedat joi graph of G[Also referred to as G coroa K i [3]] Operatio [9] Itroduce isolated vertices u i, i = to ad joi u i to the eighbors of v i The resultat graph is called the splittig graph of G Operatio 3 I additio to G itroduce two sets of isolated vertices U = {u i } ad W = {w i } correspodig to V = {v i }, i = to Joi u i ad w i to the eighbors of v i ad the w i to the vertices i U correspodig to the eighbors of v i i G for each i = to The resultat graph is called the double splittig graph of G Operatio 4 I additio to G itroduce two more copies of G o U = {u i } ad W = {w i } correspodig to V = {v i }, i = to Joi u i to the eighbors of v i ad the w i to u i for each i = to The resultat graph is called the compositio graph of G Operatio 5 I additio to G itroduce two more copies of G o U = {u i } ad W = {w i } correspodig to V = {v i }, i = to Joi w i to the eighbors of v i ad vertices i U correspodig to the eighbors of v i i G for each i = to Lemma Let G be a graph o vertices with specg) = {,, } ad H i be the graph obtaied from Operatio i, i = to 5 The i ± i spech ) = + 4 { ± ) } 5 spech ) = i { spech 3 ) = i, ± ) } i { } spech 4 ) = i, i ± i + { spech 5 ) = i, ± ) } i Proof The proof follows from Table which gives the adjacecy matrix of H i s for i = to 5 ad its spectrum, obtaied usig Lemma ad the spectrum of tesor product of matrices Table

3 38 G Idulal et al / Reciprocal Graphs Graph Adjacecy matrix Spectrum [ ] { } A I H i ± i +4 I 0 [ ] [ ] A A { ) } H = A ± 5 A 0 0 i A A A H 3 A 0 A = A 0 { i, ± ) } i A A 0 0 H 4 H 5 A A 0 A A I 0 I A A 0 A 0 A A = A A A A 0 0 { } i, i ± i + { i, ± ) } i Note: H 3 = H 5 whe G is bipartite Theorem The pedat joi graph of a graph G is reciprocal if ad oly if G is bipartite Proof Let G be a bipartite graph ad H, its pedat joi graph The, correspodig to a o-zero eigevalue of G, is also a eigevalue of G [4] By Lemma, spech) = { ± +4, specg)} Let α = + +4 be a eigevalue of H The α = ) + 4 = + ) + 4 ) + 4 = ) = ) + ) + 4 is a eigevalue of H as is a eigevalue of G Similarly for α = +4 also The eigevalues of H correspodig to the zero eigevalues of G if ay, are ad which are self reciprocal Therefore H is a reciprocal graph The coverse ca be proved by retracig the argumet Note This theorem elarges the classes of reciprocal graphs metioed i [0] The claim i [0] that the pedat joi graph of C is reciprocal for every is ot correct as C is ot bipartite for odd Defiitio A graph G is partially reciprocal if Examples:- Pedat joi graph of ay graph Splittig graph of ay reciprocal graph specg) for every specg) Theorem The splittig graph of G is reciprocal if ad oly if G is partially reciprocal Proof Let G be partially reciprocal ad H be its splittig graph Let α spech) The by Lemma 3, α = ) ) ± 5, specg) Without loss of geerality, take α = + 5 The α ) = 5 Thus α spech) as G is partially reciprocal ad hece H is reciprocal Coversely assume that H is reciprocal The by the structure of spech) as give by Lemma, G is partially reciprocal

4 G Idulal et al / Reciprocal Graphs 383 Theorem 3 Let G be a reciprocal graph The the double splittig graph ad the compositio graph of G are reciprocal if ad oly if G is bipartite Proof Let G be a bipartite reciprocal graph The specg),, specg) Let H ad H respectively deote the double splittig graph ad compositio graph of G The usig Lemma ad Table it follows that H ad H are reciprocal Table SpecH) SpecH spech) ) spech ) {, ± ) } {, ± } ) {, ± + } { }, ± ) + Coverse also follows Illustratio: The followig graphs are reciprocal whe G = P 4 3 A upperboud for the eergy of reciprocal graphs The followig bouds o the eergy of a graph are kow [5] m + ) det A EG) m ) [8] EG) m + ) m 4 m ) 3 [9] EG) 4m + ) m 8 m, if G is bipartite I this sectio we derive a better upperboud for the eergy of a reciprocal graph ad prove that the boud is best possible A graph of order ad size m is referred to as a, m) graph m+) Theorem 34 Let G be a, m) reciprocal graph The EG) ad the boud is best possible for G = tk ad tp 4 Proof Let G be a, m) reciprocal graph with specg) = {,, } Therefore i = i = E ad i = i = m

5 384 G Idulal et al / Reciprocal Graphs Now we have []the followig iequality for real sequeces a i, b i ad c i, i a i c i b i c i a i b i + a i ) / bi ) / Takig a i = i, b i = i ad c i = i =,,,, we have [EG)] [ + m] ad hece EG) m+) Whe G = tk, = t, m = t, EG) = t ad whe G = tp 4, = 4t, m = 3t, EG) = t 5 c i 4 Equieergetic reciprocal graphs I this sectio we prove the existece of a pair of equieergetic reciprocal graphs o every = p ad = 6p, p 3 Theorem 45 Let G be K p ad F be the graph obtaied by applyig Operatios 3, ad o G ad F, the graph obtaied by applyig Operatios 5, ad o G successively The F ad F are reciprocal ad equieergetic o p vertices ) p Proof Let G = K p We have speck p ) = p Let G 3 be the graph obtaied by applyig Operatio 3 o G The by Lemma, p ) ± ) p ) ± ) ) specg 3 ) = p each oce each p times Now, let G 3 be the graph obtaied by applyig Operatio o G 3 The by Lemma specg 3 ) = p ± p ) +4 ± 5 + )p )± {+ )p )} +4 each oce each p times each oce )p )± { )p )} +4 + )± {+ )} +4 )± { )} +4 each oce each p times each p times The EG 3 ) = p ) { 5 p ) + + ) p )} + 4 { + [ ) p )} p ) + ) ] ) + 4 = p ) p ) + p ) p ) p ) p ) + 6 Now, let F be the graph obtaied by applyig Operatio o G 3 The by Lemma, EF ) = 5EG 3 ) Let G 5 be the graph obtaied by applyig Operatios 5 ad o G successively ad F be that obtaied by applyig Operatio o G 5 The we have EF ) = 5EG 5 ) = 5EG 3 ) = EF ) Also by Theorem, F ad F are reciprocal Thus the theorem follows Lemma 43 Let G be a o-bipartite graph o p vertices with specg) = {,, p } ad a adjacecy matrix A The the spectra of graphs whose adjacecy matrices are

6 G Idulal et al / Reciprocal Graphs 385 F = { i, i, A A A A A A 0 A A 0 A A A A A 0 3± ad H = ) i } p ad { i, i, 0 A A A A 0 A A A A A A A A A 0 3± are ) i } p respectively Theorem 46 Let G be K p Let T ad T be the graphs obtaied by applyig Operatios ad successively o graphs associated with F ad H respectively The T ad T are reciprocal ad equieergetic o 6p vertices Proof Let the graph associated with F be also deoted by F ad F, Operatio o F The by a similar computatio as i Theorem 5, the graph obtaied by applyig EF p ) = ) p ) ) p ) ) p ) p ) + 3 ) ) + 4 ad ET ) = 5EF ) = 5EH ) = ET ), by Lemma Also by Theorem, T ad T are reciprocal Hece the theorem 5 Wieer idex of some reciprocal graphs I this sectio we derive the Wieer idices of some classes of reciprocal graphs described i the earlier sectio We shall deote by DG) = D, the distace matrix of G ad d i, the sum of etries i the i th row of D The followig theorem geeralizes the results i [4] Theorem 57 Let G be a graph with Wieer idex WG) Let H be the pedat joi graph of G The WH) = 4WG) + ) Proof We have, WG) = d i Let VG) = {v, v,, v } ad let U = {u, u,, u } be the correspodig vertices used i the pedat joi of G The the distace matrix of H is as follows 0 dv, v ) dv, v ) + dv, v ) + dv, v ) dv, v ) 0 + dv, v ) + dv, v ) + dv, v ) 0 + dv, v ) + dv, v ) + dv, v ) + dv, v ) 0 sice dv i, u j ) = ; if i = j = + dv i, v j ); i = jad du i, u j ) = du i, v i ) + dv i, v j ) + dv j, u j ) = + dv i, v j )

7 386 G Idulal et al / Reciprocal Graphs The row sum matrix of H is d + d + d + 3 d + 3 The WH) = [ d i + ) + d i + 3 ) = 4WG) + ) Hece the theorem ] The proof techiques of the followig theorems are o similar lies Theorem 58 Let G be a triagle free, m) graph ad H, its splittig graph The WH) = 4WG) + m + ) Corollory 5 Let G be a triagle free, m) graph ad F, the splittig graph of the pedat joi graph of G The WF) = [8WG) m + )] Theorem 59 Let G be a triagle free, m) graph ad H, its double splittig graph The WH) = 9WG) + 4m + 6 Theorem 50 Let G be a triagle free, m) graph ad H, its compositio graph The WH) = 9WG) Refereces [] R Balakrisha, The eergy of a graph, Li Algebra Appl, ), [] K Balińska, DM Cvetković, Z Radosavljević, S Simić, D Stevaović, A Survey o itegral graphs, Uiv Beograd Publ Elektroteh Fak Ser Mat, 00), 4 65 [3] S Barik, S Pati, BK Sarma, The spectrum of the coroa of two graphs, SIAM J Discrete Math, 007), [4] DM Cvetkovi?, M Doob, H Sachs, Spectra of Graphs-Theory ad Applicatios, Academic Press, 980) [5] DM Cvetković, I Gutma, O spectral structure of graphs havig the maximal eigevalue ot greater tha two, Publ Ist Math, 8975), [6] JR Dias, Properties ad relatioships of right-had mirror-plae fragmets ad their eigevectors : the cocept of complemetarity of molecular graphs, Mol Phys, ), [7] JR Dias, Properties ad Relatioships of Cojugated Polyees Havig a Reciprocal Eigevalue Spectrum Dedralee ad Radialee Hydrocarbos, Cro ChemActa, ), [8] J Koole, V Moulto, Maximal eergy graphs, AdvApplMath, 600), 47 5 [9] J Koole, V Moulto, Maximal eergy bipartite graphs, Graphs ad Combi, 9003), 5 [0] G Idulal, A Vijayakumar, O a pair of equieergetic graphs, MATCH Commu Math Comput Chem, 55006), [] G Idulal, A Vijayakumar, Eergies of some o-regular graphs, J Math Chem 4 007), [] G Idulal, A Vijayakumar, Some ew itegral graphs, Applicable Aalysis ad Discrete Mathematics, 007), 40 46

8 G Idulal et al / Reciprocal Graphs 387 [] G Idulal, A Vijayakumar, Equieergetic self-complemetary graphs, Czechoslovak Math J ), 9 99 [4] B Madal, M Baerjee, AK Mukherjee, Wieer ad Hosoya idices of reciprocal graphs, Mol Phys, 03005), [5] BJ McClellad, Properties of the latet roots of a matrix:the estimatio of π electro eergy, J Chem Phys, 54)97), [6] S Nikolić, N Triajstić, M Radić, Wieer idex revisited, Chem Phys Lett, 3300), 39 3 [7] HS Ramae, HB Walikar, SB Rao, BD Acharya, I Gutma, PR Hampiholi, SR Jog, Equieergetic graphs, Krajugevac J Math, 6004), 5 [8] M Radić, X Guo, T Oxley, HK Krishapriya, Wieer Matrix:Source of ovel graph ivariets, J Chem If Comp Sci, 335)993), [9] E Sampathkumar, HB Walikar, O the splittig graph of a graph, Karatak Uiv J Sci, 35/ ), 6 [0] J Sarkar, AK Mukherjee, Graphs with reciprocal pairs of eigevalues, Mol Phys, 90997), [] JM Steele, The Cauchy-Schwarz Master Class, Cambridge Uiversity Press 004) [] D Stevaović, Whe is NEPS of graphs coected?, Liear Algebra Appl, 30999), 7 44 [3] D Stevaović,Eergy ad NEPS of graphs, Liear Multiliear Algebra, 53005), Received: November, 05; Accepted: March 5, 06 UNIVERSITY PRESS Website: