E. R. van Dam and W. H. Haemers [15] conjectured that almost all graphs are determined by their spectra. Nevertheless, the set of graphs which are known to be determined by their spectra is small. Hence, discovering infinite classes of graphs that are determined by their spectra can be an interesting problem. The aim of this paper is to characterize new classes of multicone graphs that are determined by their spectrum. A multicone graph is defined to be the join of a clique and a regular graph. It is proved that any graph cospectral with multicone graph Kw 5 L(P) is determined by its adjacency spectrum as well as its Laplacian spectrum, where Kw and L(P) denote a complete graph on w vertices and the line graph of the Petersen graph, respectively. Finally, three problems for further researches are proposed.Publisher's Versio

Abdian, Ali Zeydi

TWMS Journal of Applied and Engineering Mathematics

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Isik University Academic Open Access

TWMS J. App. Eng. Math. V.7, N.1, 2017, pp. 181-187GRAPHS COSPECTRAL WITH MULTICONE GRAPHS Kw 5 L(P )ALI ZEYDI ABDIAN1, §Abstract. E. R. van Dam and W. H. Haemers [15] conjectured that almost all graphsare determined by their spectra. Nevertheless, the set of graphs which are known tobe determined by their spectra is small. Hence, discovering infinite classes of graphsthat are determined by their spectra can be an interesting problem. The aim of thispaper is to characterize new classes of multicone graphs that are determined by theirspectrum. A multicone graph is defined to be the join of a clique and a regular graph. Itis proved that any graph cospectral with multicone graph Kw 5 L(P ) is determined byits adjacency spectrum as well as its Laplacian spectrum, where Kw and L(P ) denotea complete graph on w vertices and the line graph of the Petersen graph, respectively.Finally, three problems for further researches are proposed.Keywords: adjacency spectrum, Laplacian spectrum, DS graph, line graph of Petersengraph.AMS Subject Classification: 05C50.1. IntroductionAll graphs considered here are simple and undirected. All terminology and notations ongraphs not defined here, can be found in [3, 4, 5, 19]. Let G be a graph with the adjacencymatrix A. We denote det(λI − A), the characteristic polynomial of G, by PG(λ). Themultiset of eigenvalues of A is called the adjacency spectrum, or simply the spectrum of G.Since A is a symmetric matrix, the eigenvalues of G are real. Two non-isomorphic graphswith the same spectrum are called cospectral. We say that a graph is determined by thespectrum (DS for short) if there is no other non-isomorphic graph with the same spectrum.For two graphs G1 and G2 with disjoint vertex sets V (G1) and V (G2) and disjoint edgesets E(G1) and E(G2) the disjointunion of G1 and G2 is the graph G = G1 ∪G2 with thevertex set V (G1) ∪ V (G2) and the edge set E(G1) ∪ E(G2). The join G5 H of simpleundirected graphs G and H is the graph with the vertex set V (G5H) = V (G)∪V (H) andthe edge set E(G5H) = E(G) ∪E(H) ∪ {uv|u ∈ V (G), v ∈ V (H)}. For a graph G, letD(G) = diag(d1, d2, ..., dn) be the diagonal matrix of vertex degrees, and let A(G) be theadjacency matrix of G. The matrices SL(G) = D(G) +A(G) and Lp(G) = D(G)−A(G)are called the signless Laplacian matrix and Laplacian matrix of G, respectively. Linegraph of a graph G denoted by L(G), is a graph that its vertices are edges of G and twovertices of L(G) are adjacent if their corresponding edges in G have a common vertex.The background of the question ” Which graphs are determined by their spectrum?”1 Department of Mathematics, College of Science, Lorestan University, Lorestan, Khorramabad, Iran.e-mail: azeydiabdi@gmail.com, aabdian67@gmail.com;ORCID: http://orcid.org/0000-0002-3637-2952§ Manuscript received: June 12, 2016; accepted: November 02, 2016.TWMS Journal of Applied and Engineering Mathematics, Vol.7, No.1; c© Işık University, Departmentof Mathematics, 2017; all rights reserved.181182 TWMS J. APP. ENG. MATH. V.7, N.1, 2017originates from Chemistry (in 1956, Günthadr and Primas [8] raised this question in thecontext of Hückel’s theory). A remarkable fact is that there are many wonderful papers tostudying cospectral graphs and introducing kinds of methods of constructing them (see [7],[9], [10], [11], [17] and [18], for example). In [15], it is conjectured that almost all graphsare DS. Nevertheless, the set of graphs which are known to be DS is small and thereforeit would be interesting to find more examples of DS graphs. For a survey of the subject,see [15, 16]. Special classes of multicone graphs were characterizd in [17]. In [17], Authorsinvestigated on the spectral characterization of multicone graphs and also they claimedthat friendship graph Fn is DS with respect to its adjacency spectra. In [1], Abdian andMirafzal characterized new classes of multicone graphs that were DS with respect to theirspectra. In this paper, we characterize another new classes of multicone graphs that areDS with respect to their spectra.2. PreliminariesIn this section, we give some results that will be used in the sequal.Lemma 2.1. [2, 13] Let G be a graph. For the adjacency matrix and Laplacian matrix,the following can be obtained from the spectrum:(i) The number of vertices,(ii) The number of edges.For the adjacency matrix, the following follows from the spectrum:(iii) The number of closed walks of any length.(iv) Being regular or not and the degree of regularity.(v) Being bipartite or not.For the Laplacian matrix, the following follows from the spectrum:(vi) The number of components.Theorem 2.1. [5] If G1 is r1-regular with n1 vertices, and G2 is r2-regular with n2 vertices,then the characteristic polynomial of the join G1 5G2 is given by:PG15G2(x) =PG1(x)PG2(x)(x− r1)(x− r2)((x− r1)(x− r2)− n1n2).Theorem 2.2. [1] Let G be a simple graph with n vertices and m edges. Let δ = δ(G)be the minimum degree of vertices of G and %(G) be the spectral radius of the adjacencymatrix of G. Then%(G) ≤ δ − 12+√2m− nδ + (δ + 1)24.Equality holds if and only if G is either a regular graph or a bidegreed graph in which eachvertex is of degree either δ or n− 1.Theorem 2.3. [12] Let G and H be two graphs with Laplacian spectrum λ1 ≥ λ2 ≥ ... ≥ λnand µ1 ≥ µ2 ≥ ... ≥ µm, respectively. Then Laplacian spectra of G and G 5 H aren − λ1, n − λ2, ..., n − λn−1, 0 and n + m,m + λ1, ...,m + λn−1, n + µ1, ..., n + µm−1, 0,respectively.Lemma 2.2. [12] Let G be a graph on n vertices. Then n is Laplacian eigenvalue of G ifand only if G is the join of two graphs.Proposition 2.1. [1] For a graph G, the following statements are equivalent:(i) G is d-regular.(ii) %(G) = dG, the average vertex degree.(iii) G has v = (1, 1, ..., 1)t as an eigenvector for %(G).ALI ZEYDI ABDIAN: GRAPHS COSPECTRAL WITH MULTICONE... 183Theorem 2.4. [5, 14] Let G− j be the graph obtained from G by deleting the vertex j andall edges containing j. Then PG−j(x) = PG(x)m∑i=1α2ijx− µi, where m, αij and PG are thenumber of distinct eigenvalues of graph G, main angle of G and characteristic polynomialof G, respectively.The rest of this paper is organized as follows. In Section 3, we prove that multiconegraphs Kw 5 L(P ) are determined by their adjacency spectrum. In Section 4, we showthat these graphs are DS with respect to their Laplacian spectrum. In Section 5, wereview what was said in the previous sections and finally we propose three conjecturesfor further researches.Table 1. The line graph of the Petersen graph and some its graphical propertiesThe line graph of the Petersen graph Some graphical properties|V | 15|E| 30Girth 3Graphical properties Regular, Non-bipartite graph.Adjacency spectrum of line graph of the Perersen graph{[4]1, [2]5 , [−1]4, [−2]5}3. Main results3.0.1. Graphs cospectral with multicone graphs Kw 5 L(P ).In this subsection, we show that any graph cospectral with a multicone graphKw5L(P )is determined by its adjacency spectrum.Proposition 3.1. Let G be a graph. If Spec(G) = Spec(Kw 5 L(P )). Then Spec(G) =[−1]w−1, [2]5 , [−1]4, [−2]5,[Ω +√Ω2 + 4Γ2]1,[Ω−√Ω2 + 4Γ2]1, where Ω = w + 3and Γ = 11w + 4.Proof. By Theorem 2.2 and Spec(L(P )) ={[4]1, [2]5 , [−1]4, [−2]5}the proof is com-pleted. In Lemma 3.3 we show that any graph cospectral with a multicone graph Kw 5 L(P )must be bidegreed.Lemma 3.1. Let G be cospectral with a multicone graph Kw5L(P ). Then δ(G) = w+4.Proof. Assume δ(G) = w + 4 + x, where x is an integer number. First, it is clear that inthis case the equality in Theorem 2.3 happens, if and only if x = 0. We show that x = 0.By contrary, we suppose that x 6= 0. It follows from Theorem 2.3 and Proposition 3.1 that%(G) =w + 3 +√8k − 4l(w + 4) + (w + 5)22<w + 3 + x+√8k − 4l(w + 4) + (w + 5)2 + x2 + (2w + 10− 4l)x2, where the integer num-bers k and l denote the number of edges and the number of vertices of the graph G,respectively. A simple computation follows that√Q−√Q+ g(x) < x.184 TWMS J. APP. ENG. MATH. V.7, N.1, 2017, where Q = 8k − 4l(w + 4) + (w + 5)2 ≥ 0 and g(x) = x2 + (2w + 10 − 4l)x. In thefollowing, we denote the absolute value of y by |y|, where y is an arbitrary integer number.Now, we consider two cases:Case 1. x < 0.|√Q −√Q+ g(x)| > |x| =⇒ 2Q + g(x) − 2√Q(Q+ g(x)) > x2 =⇒ Q + (w + 5 −2l)x >√Q(Q+ g(x) =⇒ Q2 + [(w + 5 − 2l)x]2 + 2Q(w + 5 − 2l)x > Q2 + Qg(x) =⇒[(w + 5 − 2l)x]2 + 2Q(w + 5 − 2l)x > Qg(x) =⇒ [(w + 5 − 2l)x]2 + 2Q(w + 5 − 2l)x >Q(x2+(2w+10−4l)x) =⇒ [(w+5−2l)x]2+Q(2w+10−4l)x > Q(x2+(2w+10−4l)x) =⇒(w + 5− 2l)2 > Q =⇒ (w + 5− 2l)2 > 8k − 4l(w + 4) + (w + 5)2 =⇒ k < l(l − 1)2.Therefore, if x < 0, then G cannot be a complete graph. In other words, if G is acomplete graph, then x > 0. Or one can say that for a complete graph G cospectral witha multicone graph Kw 5 L(P ), we must have δ(G) > w + 4.Case 2. x > 0.In the same way of Case 1, one can conclude that for a complete graph G cospectralwith a multicone graph Kw 5 L(P ), we must have δ(G) < w + 4.But, Case 1 and Case 2 are contradict to each other. So, we must have x = 0. Therefore,the assertion holds. Lemma 3.2. If G be a graph and Spec(G) = Spec(Kw 5 L(P )), then G is bidegreed, inwhich any vertex of G is of degree w + 4 or w + 14.Proof. Lemma 3.2 together with Theorem 2.3 and Proposition 2.6 complete the proof. Figure 1. Petersen graphLemma 3.3. Any graph cospectral with the multicone graph K15L(P ) is DS with respectto its adjacency spectrum.ALI ZEYDI ABDIAN: GRAPHS COSPECTRAL WITH MULTICONE... 185Proof. By Lemma 3.3, G has one vertex of degree 15, say j. On the other hand, Proposition2.7, follows that PG−j(λ) = (λ−µ3)4(λ−µ4)3(λ−µ5)4[α21jA+α22jB+α23jC+α24jD+α25jE].A = (λ− µ2)(λ− µ3)(λ− µ4)(λ− µ5),B = (λ− µ1)(λ− µ3)(λ− µ4)(λ− µ5),C = (λ− µ1)(λ− µ2)(λ− µ4)(λ− µ5),D = (λ− µ1)(λ− µ2)(λ− µ3)(λ− µ5),E = (λ− µ1)(λ− µ2)(λ− µ3)(λ− µ4),where µ1 =4 +√762, µ2 =4−√762, µ3 = 2, µ4 = −1 and µ5 = −2.We know that G− j has 15 eigenvalues. Also, if we remove the vertex j of graph G, thenthe number of edges and triangles that are removed of graph G is 15 and 30,respectively. Moreover, Lemma 3.3 follows that G− j is regular and degree of itsregularity is 4. By Lemma 2.1 (iii) for the closed walks of lengths 1, 2 and 3 we have:x+ y + z + 4 = −(4µ3 + 3µ4 + 4µ5)x2 + y2 + z2 + 16 = 60− (4µ23 + 3µ24 + 4µ25)x3 + y3 + z3 + 64 = 60− (4µ33 + 3µ34 + 4µ35), where x, y and z are eigenvalues of PG−j(λ). If we solve the above equations, thenx = 2, y = −1 and z = −2. So, Spec(G− j) ={[4]1 , [2]5, [−1]4, [−2]5}. It is well-knownthe line graph of the Petrsen graph is DS with respect to its adjacency spectrum.Therefore, G− j ∼= L(P ) and so G ∼= K1 5 L(P ).Up to now, we have shown the multicone graph K1 5 L(P ) is DS. The natural questionsis; what happen for multicone graphs Kw 5 L(P )? we answer to this question in thenext theorem.Theorem 3.1. Any graph cospectral with a multicone graphs Kw 5 L(P ) is isomorphicto Kw 5 L(P ).Proof. We solve the problem by induction on w. If w = 1, there is nothing for proof. Letthe problem be true for w; that is, if Spec(G1) = Spec(Kw5L(P )), then G1 ∼= Kw5L(P ),where G1 is an arbitrary graph cospectral with multicone graph Spec(G) = Spec(Kw 5L(P )). We show that Spec(G) = Spec(Kw+1 5 L(P )) follows that Kw 5 L(P ), where Gis an arbitrary graph cospectral with multicone graph Spec(G) = Spec(Kw+1 5 L(P )).First, it is obvious that G has one vertex and w+15 edges more than G1. Now, by Lemma3.3, it follows that G1 has w vertices of degree w+ 14 and 15 vertices of degree w+ 4 andalso G has w + 1 vertices of degree w + 15 and 15 vertices of degree w + 5. So, we musthave G ∼= K1 5G1. Now, the induction hypothesis completes the proof. In the following, we present another proof of Theorem 3.5.Proof. Let Spec(G) = Spec(Kw 5 L(P )). By Lemma 3.3, G has graph Γ as its subgraphin which degree of any vertex of Γ is w+ 14. On the other words, G ∼= Kw5H, where His a subgraph of G. Now, we remove vertices of Kw and we consider 15 another vertices.Degree of graph consists of these vertices is 4, say H. By Theorem 2.2, Spec(H) ={[4]1, [2]5 , [−1]4, [−2]5}= Spec(L(P )). This completes the proof. In the following, we show that multicone graphs Kw 5 L(P ) are DS with respect to theiradjacency spectrum.186 TWMS J. APP. ENG. MATH. V.7, N.1, 20174. Connected graphs cospectral with a multicone graph Kw 5 L(P ) withrespect to Laplacian spectrumTheorem 4.1. Multicone graphs Kw5L(P ) are DS with respect to their Laplacian spec-trum.Proof. We solve the problem by induction on w. If w = 1, there is nothing to prove. Letthe problem be true for values of less than w; that is, Spec(Lp(G1)) = Spec(Lp(Kw 5L(P )) implies that G1 ∼= Kw 5 L(P ), where G1 is a graph. We show that the prob-lem is true for w + 1; that is, we show that Spec(Lp(G)) = Spec(Lp(Kw+1 5 L(P ))) ={[w + 16]w+1, [w + 3]4, [0]1, [w + 6]4, [w + 7]5}follows thatG ∼= Kw+15L(P ), whereG is agraph. But, Spec(Lp(G1)) = Spec(Lp(Kw5L(P ))) ={[w + 15]w, [w + 2]4, [0]1, [w + 5]4, [w + 6]5}implies that G1 ∼= Kw 5 L(P ). Theorem 2.5 implies that G1 and G are the join of twographs. On the other hand, Spec(Lp(K15G1)) = Spec(Lp(G)) = spec(Lp(Kw+15L(P )))and also G has one vertex and w + 15 edges more than G1. Therefore, we must haveG ∼= K1 5G1. Now, the induction hypothesis completes the proof. 5. Conclusion remark and ConjecturesNow, we review what were proved in the earlier sections and finally we pose threeconjectures.Corollary 5.1. Let G be a graph. The following statements are eguivalent:(i) G ∼= Kw 5 L(P ),(ii) Spec(G) = Spec(Kw 5 L(P )),(iii) Spec(Lp(G)) = Spec(Lp(Kw 5 L(P )).In the following, we pose three conjectures.Conjecture 5.1. Multicone graphs Kw5L(P ) are DS with respect to their signless Lapla-cian spectrum.Conjecture 5.2. The complement of multicone graphs Kw 5 L(P ) are DS with respectto their adjacency spectrum.Conjecture 5.3. The complement of multicone graphs Kw 5 L(P ) are DS with respectto their signless Laplacian spectrum.References[1] Abdian,A.Z. and Mirafzal,S.M., (2015), On new classes of multicone graphs determined by theirspectrums, Alg. Struc. Appl., 2, pp.23-34.[2] Abdollahi,A., Janbaz,S., and Oubodi,M., (2013), Graphs cospectral with a friendship graph or itscomplement, Trans. Comb., 2, pp.37-52.[3] Bapat,R.B., (2010), Graphs and matrices, New York (NY), Springer.[4] Biggs,N.L., (1933), Algebraic Graph Theory, Cambridge university press.[5] Cvetković,D., Rowlinson,P., and Simić,S., (2010), An Introduction to the Theory of graph Spectra,London Mathematical Society Student Texts, 75, Cambridge University Press.[6] Das,K.C., (2013), Proof of conjectures on adjacency eigenvalues of graphs, Disceret Math, 313 pp.19-25.[7] Godsil,C.D. and McKay,B.D., (1982), Constructing cospectral graphs, Aequationes Math., 25, pp.252-268.[8] Günthard,Hs.H. and Primas,H., (1956), Zusammenhang von Graphtheorie und Mo-Theorie von-Molekeln mit Systemen konjugierter Bindungen., Helv. Chim. Acta, 39, pp.1645-1653.[9] Haemers,W.H. and Spence,E., (2004), Enumeration of cospectral graphs, Eur. J. Comb., 25 pp.199211.ALI ZEYDI ABDIAN: GRAPHS COSPECTRAL WITH MULTICONE... 187[10] Harary,F., King,C., Mowshowitz,A., and Read,R., (1971), Cospectral graphs and digraphs, Bull. Lond.Math. Soc., 3, pp.321328.[11] Johnson,C.R. and Newman,M., (1980), A note on cospectral graphs, J. Comb. Theory, Ser. B, 28,pp.96103.[12] Mohammadian,A. and Tayfeh-Rezaie,B., (2011), Graphs with four distinct Laplacian eigenvalues, J.Algebraic Combin., 34, pp.671-682.[13] Omidi,G.R., (2009), On graphs with largest Laplacian eignnvalues at most 4, Australas. J. Combin.,44, pp.163-170.[14] Rowlinson,P., (2007), The main eigenvalues of a graph: a survey, Appl. Anal. Discrete Math., 1,pp.445-471.[15] van Dam,E.R. and Haemers,W.H., (2003), Which graphs are determined by their spectrum?, LinearAlgebra. Appl., 373, pp.241-272.[16] van Dam,E.R. and Haemers,W.H., (2009), Developments on spectral characterizations of graphs,Discrete Math., 309, pp.576-586.[17] Wang,J., Zhao,H., and Huang,Q., (2012), Spectral charactrization of multicone graphs, Czech. Math.J., 62, pp.117-126.[18] Wang,J., Belardo,F., Huang,Q., and Borovićanin,B., (2010), On the two largest Q-eigenvalues ofgraphs, Discrete Math., 310, pp.2858-2866.[19] West,D.B., (2001), Introduction to Graph Theory, Upper Saddle River, Prentice hall.Ali Zeydi Abdian is a Ph.D student of mathematics at the School of Mathematicsand Computer Science at Lorestan University. His research areas are combinatorics,algebraic graph theory, graph theory, ring theory, group theory, group automorphismof a graph, and graphs that are determined by their spectra.

Graphs cospectral with multicone graphs KW 5 L(P)

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Graphs cospectral with multicone graphs KW 5 L(P)

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