Disjoint unions of complete graphs characterized by their Laplacian spectrum

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1 Electroic Joural of Liear Algebra Volume 18 Volume 18 (009) Article Disjoit uios of complete graphs characterized by their Laplacia spectrum Romai Boulet Follow this ad additioal works at: Recommeded Citatio Boulet, Romai. (009), "Disjoit uios of complete graphs characterized by their Laplacia spectrum", Electroic Joural of Liear Algebra, Volume 18. DOI: This Article is brought to you for free ad ope access by Wyomig Scholars Repository. It has bee accepted for iclusio i Electroic Joural of Liear Algebra by a authorized editor of Wyomig Scholars Repository. For more iformatio, please cotact scholcom@uwyo.edu.

2 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society DISJOINT UNIONS OF COMPLETE GRAPHS CHARACTERIZED BY THEIR LAPLACIAN SPECTRUM ROMAIN BOULET Abstract. A disjoit uio of complete graphs is i geeral ot determied by its Laplacia spectrum. It is show i this paper that if oe oly cosiders the family of graphs without isolated vertex, the a disjoit uio of complete graphs is determied by its Laplacia spectrum withi this family. Moreover, it is show that the disjoit uio of two complete graphs with a ad b vertices, a b > 5 3 ad b>1 is determied by its Laplacia spectrum. A couter-example is give whe a b = 5 3. Key words. Graphs, Laplacia, Complete graph, Graph determied by its spectrum, Strogly regular graph. AMS subject classificatios. 05C50, 68R Itroductio ad basic results. The Laplacia of a graph G is the matrix L defied by L = D A, whered is the diagoal matrix of the degrees of G ad A is the adjacecy matrix of G. The Laplacia spectrum gives some iformatio about the structure of the graph but determiig graphs characterized by their Laplacia spectrum remais a difficult problem []. I this paper we focus o the disjoit uio of complete graphs. A complete graph o vertices is deoted by K ad the disjoit uio of the graphs G ad G is deoted by G G. The Laplacia spectrum of K k1 K k... K k is {k (k1 1) 1,k (k 1),...,k (k1 1), 0 () } but i geeral the coverse is ot true: a disjoit uio of complete graphs is ot i geeral determied by its Laplacia spectrum. For istace [], the disjoit uio of the Peterse graph with 5 isolated vertices is L-cospectral with the disjoit uio of the complete graph with five vertices ad five complete graphs with two vertices. These graphs are depicted i figure 1.1. I this paper we show i Sectio that the disjoit uio of complete graphs without isolated vertex is determied by its Laplacia spectrum i the family of graphs without isolated vertex. The i Sectio 3, we study the disjoit uio of two Received by the editors February 11, 009. Accepted for publicatio December 4, 009. Hadlig Editor: Miroslav Fiedler. Istitut de Mathématiques de Toulouse, Uiversité de Toulouse et CNRS (UMR 519), Frace (boulet@uiv-tlse.fr). 773

3 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society R. Boulet Fig The graph draw o the left is a graph o-isomorphic to a disjoit uio of complete graphs but Laplacia cospectral with the disjoit uio of complete graphs draw o the right complete graphs K a ad K b ad show that if a b > 5 3,theK a K b is determied by its Laplacia spectrum. To fix the otatio, the set of vertices of a graph G is deoted by V (G) ad the set of edges is deoted by E(G); for v V (G), d(v) deotes the degree of v. The complemet of a graph G is deoted by G. Cocerig the spectrum, Sp(G) = {µ (m1) 1, µ (m k) k } meas that µ i is m i times a eigevalue of L (the multiplicity of µ i is at least m i ; we may allow µ i = µ j for i j). We ed this itroductio with some kow results about the Laplacia spectrum ad strogly regular graphs. Theorem 1.1. [6] The multiplicity of the Laplacia eigevalue 0 is the umber of coected compoets of the graph. Theorem 1.. [4, 6] Let G be a graph o vertices whose Laplacia spectrum is µ 1 µ... µ 1 µ =0.The: 1. µ 1 1 mi{d(v),v V (G)}.. If G is ot a complete graph the µ 1 mi{d(v),v V (G)}. 3. µ 1 max{d(u)+d(v),uv E(G)}. 4. µ i µ i = E(G). 6. µ 1 1 max{d(v),v V (G)} > max{d(v),v V (G)}. Theorem 1.3. Let G be a graph o vertices, the Laplacia spectrum of G is: µ i (G) = µ i (G), 1 i 1 Corollary 1.4. Let G be a graph o vertices, we have µ 1 (G) with equality if ad oly if G is a o-coected graph.

4 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Disjoit Uios of Complete Graphs 775 Theorem 1.5. [] A complete graph is determied by its Laplacia spectrum. Defiitio 1.6. [5] A graph G is strogly regular with parameters, k, α, γ if G is ot the complete graph or the graph without edges G is k-regular Every two adjacet vertices have exactly α commo eighbors Every two o-adjacet vertices have exactly γ commo eighbors Theorem 1.7. [5] A regular coected graph is strogly regular if ad oly if it has exactly three distict adjacecy eigevalues. A strogly regular o-coected graph is the disjoit uio of r complete graphs K k+1 for a give r. Theorem 1.8. [5] Let G be a coected strogly regular graph with parameters, k, α, γ ad let k, θ, τ the eigevalues of its adjacecy matrix. The: θ = α γ + τ = α γ where =(α γ) +4(k γ) =(θ τ) Moreover, let m θ (resp. m τ ) the multiplicity of θ (resp. τ), the: ( 1)τ + k m θ = θ τ ( 1)θ + k m τ = θ τ That is: m θ = 1 ( 1 m τ = 1 ) k +( 1)(α γ) ( ) k +( 1)(α γ) 1+. Disjoit uio of complete graphs. The aim of this sectio is to show that if we cosider graphs without isolated vertex, the the disjoit uio of complete graphs is determied by its Laplacia spectrum.

5 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society R. Boulet We first state some results about disjoit uio of complete graphs (icludig isolated vertices). Propositio.1. The Laplacia spectrum of a graph G with oe ad oly oe positive Laplacia eigevalue a is {a (ra r), 0 (r+p) } ad G is isomorphic to K a K a K }{{ a K } 1 K 1 K 1. }{{} r times p times Proof. Let G be a graph with oe ad oly oe positive Laplacia eigevalue a ad let H be a coected compoet of G differet from K 1. The graph H has oe ad oly oe positive eigevalue a. If H is ot a complete graph, the by theorem 1. we have a mi{d(v),v V (G)} max{d(v),v V (G)} <a, cotradictio. As aresulth is a complete graph ad H is isomorphic to K a ad there exists r N, p N such that G is isomorphic to K a K a K }{{ a K } 1 K 1 K 1. }{{} r times p times Theorem.. There is o cospectral o-isomorphic disjoit uio of complete graphs. Proof. Let G = K k1 K k ad G = K k 1 K k,wehave = (same umber of coected compoets). If G ad G are ot isomorphic, the there exists λ N \{0, 1} such that the umber of coected compoets of G isomorphic to K λ is differet from the umber of coected compoets of G isomorphic to K λ. Therefore, the multiplicity of λ as a eigevalue of the Laplacia spectrum of G is differet from the multiplicity of λ as a eigevalue of the Laplacia spectrum of G ad so G ad G are ot cospectral. Theorem.3. Let G be a graph without isolated vertex. If the Laplacia spectrum of G is {k (k1 1) 1,k (k 1),...,k (k 1), 0 () } with k i N\{0, 1} the G is a disjoit uio of complete graphs of order k 1,...,k. Proof. The graph G has coected compoets (Theorem 1.1) G 1,...,G of order l 1,...,l.WedeotebyN the umber of vertices of G. Wehave N = l i = i=1 i=1 k i Let k j be a eigevalue of G, thereexistsi such that k j is a eigevalue of G i, so l i k j (Theorem 1.). Let G i be a coected compoet, as G does ot have isolated vertices we have l i > 1adG i possesses at least oe eigevalue differet from 0, let k j be this eigevalue, we have l i k j.

6 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Disjoit Uios of Complete Graphs 777 As a result j i : l i k j i j : l i k j We assume k 1 k... k > 0adl 1 l... l > 1. We ow show by iductio o j that k j l j, j = j = 0: we kow that there exists j such that k j l,sok l. Let j 0 > 0. We assume that j <j 0, k j l j et let us show that k j0 l j0 by cotradictio. If k j0 >l j0 the k j0 >l j, j <j 0 so k j, j j 0, caot be a eigevalue of G j,j<j 0.So (Sp(G j ) \{0}) {k (k j 1) j }. j<j 0 j<j 0 But Sp(G j ) \{0} j<j 0 = l j j 0 k j j 0 = j<j 0 j<j 0 so (Sp(G j ) \{0}) = {k (k j 1) j }. j<j 0 j<j 0 {k (k j 1) j } j<j 0 As a result k j, j < j 0 caot be a eigevalue of G j0 ad as k j0 >l j0, k j, j j 0 caot be a eigevalue of G j0. That implies that G j0 does ot have ay positive eigevalue which cotradicts that G is without isolated vertex. So k j0 l j0 which coclude this iductio. As l i = k i ad j =1,,, l j k j we have j =1,,, l j = k j. We ow show by iductio o j that Sp(G j ) \{0} = {k (k j 1) j }, j = 0,, 1. j =0. Letk r be a eigevalue of G the k r l = k ad as k r k we have k r = k.sothek 1 positive eigevalues of G are the k s. Let j 0 > 0. We assume that j <j 0 Sp(G j ) \{0} = {k (k j 1) j }. The k (k j 1) j for j<j 0 are ot eigevalues of G j0 ad as k j = l j l j0 for j j 0 the positive eigevalues of G j0 are ecessarily l j0 i.e. k (k j 0 1) j 0. By Theorem.1 we have that G i, i =1,, are the complete graphs o k i vertices.

7 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society R. Boulet 3. Disjoit uio of two complete graphs. I this sectio we cosider the disjoit uio of two complete graphs ad we aim to replace the coditio without isolated vertex of Theorem.3 (this coditio caot be deduced from the spectrum) by a coditio o the eigevalues. The spectrum of K a K b, a b>1is{a (a 1),b (b 1), 0 () }. The disjoit uio of two complete graphs is ot i geeral determied by its spectrum. Here is a couter-example: The Laplacia spectrum of the lie graph of K 6 (which is a strogly regular graph with parameters 15, 8, 4, 4) is {10 (9), 6 (5), 0}, so the Laplacia spectrum of L(K 6 ) K 1 is {10 (9), 6 (5), 0 () } which is also the spectrum of K 10 K 6. As the disjoit uio K a K a... K a is determied by its spectrum [], we assume a b The aim of this sectio is to show that a graph with Laplacia spectrum {a (a 1),b (b 1), 0 () } with a> 5 3b is the disjoit uio of two complete graphs. The paper [3] ad the thesis [1] study graphs with few eigevalues. We ca i particular metio the followig results: Theorem 3.1. [3, Theorem.1 ad Corollary.4] A k-regular coected graph with exactly two positive Laplacia eigevalues a ad b is strogly regular with parameters, k, α, γ with γ = ab. Moreover k verifies k k(a + b 1) γ + γ =0. Remark 3.. I [3], the relatio γ = ab ad the equatio k k(a + b 1) γ + γ = 0 are give i the proof of Theorem.1. Theorem 3.3. [3] Let G be a o-regular graph with Laplacia spectrum {a (a 1),b (b 1), 0} with a>b>1, a, b N. The G possesses exactly two differet degrees k 1 ad k (k 1 k ) verifyig: { k1 + k = a + b 1 k 1 k = ab ab ad k b with k = b if ad oly if G or G is ot coected. Lemma 3.4. A regular graph with Laplacia spectrum {a (a 1),b (b 1), 0} is a +1 strogly regular graph with parameters,, +1 4, Moreover we have (a b) = a + b. Proof. Let G be a regular graph with Laplacia spectrum {a (a 1),b (b 1), 0}, the accordig to Theorem 3.1 G is strogly regular with parameters, k, α, γ. The spectrum of the adjacecy matrix of G is {(k a) (a 1), (k b) (b 1),k}. By Theorem 1.8 we have k b = α γ+ ad k a = α γ where = (α γ) +4(k γ) =(a b) ad we have α γ =k a b so (remid that a + b 1=): (3.1) α γ =k 1 Moreover Theorem 1.8 gives b 1 = 1 ( ) 1 k+( 1)(α γ) ad

8 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society a 1= 1 Disjoit Uios of Complete Graphs 779 ( ) 1+ k+( 1)(α γ) so a b = k+( 1)(α γ) ad so (3.) (a b) =k +( 1)(α γ) But ab = γ (Theorem 3.1) so (3.3) (a b) =k + α ab (α γ) Equatios 3.1 ad 3.3 give: (3.4) (a b) =1+(α +1) ab As the mea of the degrees is k, wehavek = E Laplacia eigevalues, so k = a(a 1)+b(b 1) i.e. (3.5) k = a + b 1 ad E is the sum of the Equatio 3.4 gives a + b ab 1 = α, usig Equatio 3.5 we have k ab = α i.e. (k α) =ab. Butab = γ so (3.6) α + γ = k Usig = (α γ) +4(k γ)ad =(a b) =k+( 1)(α γ) (Equatio 3.) we obtai (α γ) +4(k γ)=k+( 1)(α γ) but 1 = α+γ+k (Equatio 3.1) ad k =α+γ (Equatio 3.6), so (α γ) +4α =α+γ +(α+3γ )(α γ) that is (3.7) (α γ)(4 4γ) =0 As a result, we have α = γ or γ =1. Letusshowthatγ = 1 is impossible: γ =1 implies α = k 1 ad Equatio 3.1 becomes = k +1 ad so G is the complete graph with vertices, which is impossible because {a (a 1),b (b 1), 0} with a>b>1isot the spectrum of a complete graph. Fially α = γ = k +1 (Equatio 3.6) ad (Equatio 3.1) k = ad (Equatio 3.) (a b) =k = +1=a + b. Theorem 3.5. Let G be a graph whose Laplacia spectrum is {a (a 1),b (b 1), 0} with a>b>1, a, b N \{0, 1} ad a> 5 3b.TheGis ot regular. Proof. Proof by cotradictio. Let G be a graph whose Laplacia spectrum is {a (a 1),b (b 1), 0} with a>b>1, a, b N \{0, 1} ad a> 5 3b ad we assume that G

9 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society R. Boulet is a k-regular graph. The by the previous lemma we have that G is strogly regular ad (a b) = a + b. This implies a ab a = b b < 0soa b 1 < 0ada b. The 3b a + b =(a b) > 4 9 b so b( 4 9 b 3) < 0whichgivesb 6. We have 5 3b<a b ad b 6, by listig the differet cases we show that (a b) a + b for b 3: b a a + b (a b) If b =3thea =6ad +1=a + b = 9 but, accordig to Lemma 3.4, + 1 is eve, a cotradictio. Remark 3.6. Whe b<a 5 3 b, the equatio (a b) = a + b admits a ifiity of solutios; it is ot difficult to show that the couples (a, b) =(u i,u i 1 )whereu 0 =3 ad u i = u i 1 + i + are solutios. Lemma 3.7. There is o graph with Laplacia spectrum {a (a 1),b (b 1), 0} with a, b N \{0, 1} ad a>b. Proof. Let G be a graph with Laplacia spectrum {a (a 1),b (b 1), 0} with a, b N \{0, 1} ad a>b, theg is ot regular (Theorem 3.5) ad by Theorem 3.3 we have that G possesses exactly two differet degrees k 1 ad k verifyig: { k1 + k = a + b 1 k 1 k = ab ab with k b + 1 because G ad G are coected (G is discoected if ad oly if the greatest eigevalue of G is G, but here G = a + b 1 a). We have ab N, but ab because a + b = +1ada, b ab a + b. So ab. The itegers k 1 ad k are solutios of the equatio x (a+b 1)x+ab ab =0whose discrimiat is = (a + b 1) 4ab +4 ab O oe had we have: ad so k 1 = a+b 1+,k = a+b 1. =(a + b) (a + b)+1 4ab +4 ab =(a b) (a + b)+1+4 ab (a b) (a + b)+9

10 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Disjoit Uios of Complete Graphs 781 ad o the other had we have (remid that k b +1): =(a + b 1 k ) (a + b 1 (b +1)) Cotradictio. (a b 3) =(a b) 6a +6b +9 =(a b) a 4a +6b + 9 but a>b i.e. 4a < 8b < (a b) a b +9=(a b) (a + b)+9 Theorem 3.8. There is o graph with Laplacia spectrum {a (a 1),b (b 1), 0} with a, b N \{0, 1} ad 5 3 b<a. Proof. Let G be a graph with Laplacia spectrum {a (a 1),b (b 1), 0} with a, b N \{0, 1} ad 5 3b<a. By Lemma 3.7 we ca assume a b. The graph G is ot regular (Theorem 3.5) ad by Theorem 3.3 we have that G possesses exactly two differet degrees k 1 ad k verifyig: { k1 + k = a + b 1 k 1 k = ab ab with k b +1. First we show that we have b 6; for that aim we use the relatio 5 3b<a b ad list the possible values of a if b<6adweshowthat does ot divide ab. This is summed up ito the followig table. b a ab Does divide ab? o o o o o Heceforth we assume b 6. Let us show that the case k = b + 1 is impossible. If k = b +1thek 1 = a. We deote by 1 (resp. ) the umber of vertices of degree k 1 (resp. k ). The sum of the degrees is o oe had k k ad o the other had a(a 1)+b(b 1) (sum of the eigevalues) i.e. k 1 (k 1 +3)+k (k 3)+4. So k 1 1 +k = k 1 (k 1 +3)+k (k 3)+4 i.e. k 1 ( 1 k 1 3)+k ( k +3) = 4. But ( 1 k 1 3)+( k +3) = 0 because = k 1 +k = 1 + so ( 1 k 1 3)(k 1 k )=4 i.e. ( 1 k 1 3)(a b 3) = 4. As a result a b 3 divides 4. If a b 3=1thea = b + 4 but a> 5 3 b = b + 3b b + 4 (because b 6). This case is impossible.

11 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society 78 R. Boulet If a b 3=thea = b +5ada> 5 3 b = b + 3b b + 5 as soo as b 8. If b =6thea =11ad = 16, ab = 66 ad 16 does ot divide 66. This case is impossible. If b =7thea =1ad = 18, ab = 84 ad 18 does ot divide 84. This case is impossible. If a b 3=4thea = b +7 ad a> 5 3 b = b + 3b b + 7 as soo as b 11. The cases 6 b 10 are cosidered i the followig table: b a ab Does divide ab? o o o yes o For the case b =9,a =16wehavek 1 =14adk = 10, k 1 k = 140 ad ab ab = 138, this case is impossible. As a result we have k b +. We have that k 1 ad k are solutios of the equatio x (a+b 1)x+ab ab =0 whose discrimiat is = (a + b 1) 4ab +4 ab. ab Let d be the mea degree of G, wehaved = a(a 1)+b(b 1) ab + a + b so = ab+(a+b)(a+b 1) = = 1 (b + a d) 4 because b 6ada>k 1 > d gives a d. We have o oe had: =(a + b) (a + b)+1 4ab +4 ab =(a b) (a + b)+1+4 ab (a b) (a + b)+17 ad o the other had (remid that a + b 1 k = k > 0): =(a + b 1 k ) (a + b 1 (b +)) Cotradictio. (a b 5) =(a b) 10a +10b +5 =(a b) (a + b) 8a +1b + 5 but 8a < 40 3 b < (a b) (a + b) 4 3 b +5 (a b) (a + b)+17. { Remark 3.9. If b < a < 5 3 b the the system k1 + k = a + b 1 k 1 k = ab ab with k 1,k,a,b, N admits solutios. If (a b) = a + b the = 1 ad the system admits a solutio if a+b is eve. The followig table shows solutios with a, b 1000 ad (a b) a + b.

12 Electroic Joural of Liear Algebra ISSN A publicatio of the Iteratioal Liear Algebra Society Disjoit Uios of Complete Graphs 783 a b k1 k Theorem The graph K a K b with a, b N\{0, 1} ad 5 3b<ais determied by its Laplacia spectrum. Proof. LetG be a graph with Laplacia spectrum {a (a 1),b (b 1), 0 () } with a, b N ad 5 3b<athe G has two coected compoets. If G has a isolated vertex the the Laplacia spectrum of a coected compoet of G is {a (a 1),b (b 1), 0}, which is impossible (Theorem 3.8). Cosequetly G does ot have isolated vertex ad we apply Theorem The followig corollary is straightforward thaks to Theorem 1.3: Corollary The complete bipartite graph K a,b with a, b N \{0, 1} ad b<ais determied by its Laplacia spectrum. REFERENCES [1] E.R. va Dam. Graphs with few eigevalues A iterplay betwee combiatorics ad algebra. PhD thesis, Tilburg Uiversity, [] E.R. va Dam ad W.H. Haemers. Which graphs are determied by their spectrum? Liear Algebra ad its Applicatios, 373:41 7, 003. [3] E.R. va Dam ad W.H. Haemers. Graphs with costat µ ad µ. Discrete Mathematics, 18(1-3):93 307, [4] M. Fiedler. Algebraic coectivity of graphs. Czechoslovak Mathematical Joural, 3(98):98 305, [5] C. Godsil ad G. Royle. Algebraic Graph Theory. Spriger, New York, 001. [6] B. Mohar. The Laplacia Spectrum of Graphs. Graph Theory, Combiatorics, ad Applicatios, : , 1991.