ON GRAPHS WITH THREE DISTINCT LAPLACIAN EIGENVALUES. 1 Introduction

Transcription

1 Appl. Math. J. Chiese Uiv. Ser. B 2007, 22(4): ON GRAPHS WITH THREE DISTINCT LAPLACIAN EIGENVALUES Wag Yi 1 Fa Yizheg 1 Ta Yigyig 1,2 Abstract. I this paper, a equivalet coditio of a graph G with t (2 t ) distict Laplacia eigevalues is established. By applyig this coditio to t =3,ifG is regular (ecessarily be strogly regular), a equivalet coditio of G beig Laplacia itegral is give. Also for the case of t =3,ifG is o-regular, it is foud that G has diameter 2 ad girth at most 5 if G is ot a tree. Graph G is characterized i the case of its beig triagle-free, bipartite ad petago-free. I both cases, G is Laplacia itegral. 1 Itroductio Let G =(V,E) be a simple graph with vertex set V = V (G) ={v 1,v 2,,v } ad edge set E = E(G) ={e 1,,e m }. Deote by d(v) the degree of v V i the graph G. The the Laplacia matrix of G is L(G) = D(G) A(G), where D(G) is the diagoal matrix diag{d(v 1 ),d(v 2 ),,d(v )} ad A(G) is the (0,1) adjacecy matrix of G. It is kow that L(G) is sigular ad positive semidefiite; ad its eigevalues ca be arraged as follows: λ 1 (G) λ 2 (G) λ (G) =0. The spectrum of G is defied by the multi-set S(G) ={λ 1 (G),λ 2 (G),,λ (G)}. The otio of itegral graphs was first itroduced i [1]. A graph is called itegral if all the eigevalues of its adjacecy matrix are itegers. The aalogous problem for L(G) isalso iterestig [2]. A graph G is said to be Laplacia itegral if S(G) cosists etirely of itegers. I geeral, the problem of characterizig itegral graphs seems difficult. Thus it makes sese to restrict our ivestigatios to some iterestig families of graphs. Merris [3] has show that the degree maximal graphs are Laplacia itegral. For some related results oe may refer to Received: MR Subject Classificatio: 05C50, 15A18. Keywords: Laplacia matrix, spectrum, Laplacia itegral, strogly regular graph. Digital Object Idetifier(DOI): /s z. Supported by the Ahui Provicial Natural Sciece Foudatio ( ), Natioal Natural Sciece Foudatio of Chia ( , ), NSF of Departmet of Educatio of Ahui Provice (2004kj027, 2005kj005zd), Foudatio of Ahui Istitute of Architecture ad Idustry( ) ad Foudatio of Mathematics Iovatio Team of Ahui Uiversity, ad Foudatio of Talets Group Costructio of Ahui Uiversity.

2 Wag Yi, et al. ON GRAPHS WITH THREE DISTINCT LAPLACIAN EIGENVALUES 479 [2,4]. I order to obtai ew Laplacia itegral graphs from kow oes by addig edges, Fa [5] itroduced the otio of spectral itegral variatio. Employig this otio ad related results [5,6],Fa [7],Kirklad [8] costructed ew Laplacia itegral graphs from kow oes by addig edges. Let G be a coected simple graph o vertices. The G has at least 2 distict eigevalues as G cotais oe zero eigevalue ad other ozero eigevalue. It is kow that G has exactly 2 distict Laplacia eigevalues if ad oly if G is a complete graph. I this case G is Laplacia itegral. Recetly, Fallat ad Kirklad, et. al. [9] studied a extreme class of Laplacia itegral graphs: the Laplacia itegral graphs with all eigevalues distict. Aother extreme case that a Laplacia itegral graph with 3 distict eigevalues is also iterestig. To hadle this case, we eed to discuss the graph with few distict Laplacia eigevalues. Dam ad Harmers [10] proved that a coected graph has two distict ozero Laplacia eigevalues if ad oly if it has costat μ ad μ. Note that a graph G is called to have costat μ ad μ if i the graph G every pair of o-adjacet vertices has μ commo eighbors, ad i its complemet graph G c every pair of o-adjacet vertices has μ commo eighbors. The graphs whose adjacecy matrices have few distict eigevalues i geeral have ice combiatioal properties. Oe may refer to [11-13] for some related results. I this paper, we first establish a equivalet coditio of a graph G with t (2 t ) distict Laplacia eigevalues. Applyig this coditio to t = 3, we obtai Dam ad Harmers s result [10, Theorem 2.1]; ad i additio if G is also regular (ecessarily be strogly regular), we give a equivalet coditio of G beig Laplacia itegral. Also for the case of t =3,ifG is o-regular, we fid that G has diameter 2 ad girth at most 5 ad characterize the graph G i the case of G beig triagle-free, bipartite ad petago-free. I both cases, G is Laplacia itegral. 2 Mai results ad proofs Deote by I,J respectively the idetity matrix ad the square matrix with all etries oes of appropriate sizes. Deote by 1 a colum vector with all etries oes of appropriate sizes. Lemma 2.1. Let G be a coected graph o 3 vertices ad L its Laplacia matrix. The L has t (2 t ) distict eigevalues if ad oly if there exist t 1 distict umbers μ 1,,μ t 1 such that i=1 (L μ ii) =( 1) J. (2.1) Proof. We first prove the sufficiecy. Multiplyig L ad the both sides of (2.1), we get L(L μ 1 I)(L μ 2 I) (L μ t 1 I)=0, which implies that the miimal polyomial of L is x(x μ 1 )(x μ 2 ) (x μ t 1 ), ad hece L has t distict eigevalues 0,μ 1,,μ t 1. For the ecessity, let μ 1,,μ t 1 be the ozero distict eigevalues of L. The the miimal polyomial of L is x(x μ 1 )(x μ 2 ) (x μ t 1 ), which implies that Π t 1 t 1 Πt 1 i=1 μ i LΠ t 1 i=1 (L μ ii) =0.

3 480 Appl. Math. J. Chiese Uiv. Ser. B Vol. 22, No. 4 Sice G is coected, ay eigevector of L correspodig to the zero eigevalue is a scalar multiple of the vector 1. So the ith colum vector of matrix Πi=1 t 1 (L μ ii) ca be writte i the form c i 1 for some c i for i =1,,, ad hece Π t 1 i=1 (L μ ii) =1(c 1,c 2,,c ). Multiply 1 T ad the both sides of the above equality, we have ( 1) t 1 Π t 1 i=1 μ i1 T = (c 1,c 2,,c ), ad hece for i =1, 2,,, t 1 Πt 1 i=1 c i =( 1) μ i. The result follows. Corollary 2.2. Let G be a coected graph o 3 vertices. The G has exactly two distict eigevalues if ad oly if G is a complete graph. Proof. By Lemma 2.1 G has exactly two distict eigevalues if ad oly if there exists a ozero umber r such that L ri = r J. As L is a iteger matrix ad r, sor = i the above equality. The result follows. Corollary 2.3. Let G be a coected grapho vertices. The G has three distict eigevalues if ad oly if there exist two distict positive umbers r, s such that (d i r)(d i s)+d i = rs, (2.2) (r d i )+(s d j )+ N vi N vj = rs/, for {v i,v j } E(G), (2.3) N vi N vj = rs/, for {v i,v j } / E(G). (2.4) Proof. By Lemma 2.1 G has three distict eigevalues if ad oly if there exist two distict positive umbers r, s such that (L ri)(l si) = rs J. (2.5) By cosiderig the diagoal etries (i, i) ad o-diagoal etries (i, j) for both sides of above equality, the result follows. Remark. I [10], the authors discussed a graph havig 3 distict eigevalues ad obtaied Eq. (2.5) i the proof of Theorem 2.1 of [10]. Here we provide a differet proof to get Lemma 2.1 for a graph havig t( 2) distict eigevalues. For Eq. (2.3), oe ca fid N vi N vj = r + s rs so that i the complemet graph Gc, N vi N vj = + rs (r + s), for {v i,v j } / E(G c ), where N v deotes the eighbor of vertex v i graph G c. Hece a coected graph o vertices has two ozero distict eigevalues r, s if ad oly if G has costat μ = rs rs ad μ = + (r + s), which is cosistet to Dam ad Hamers s result [10, Theorem 2.1]. Now let G be a coected simple graph o vertices with three distict eigevalues give respectively by 0,r,s (r s). By Eq.(2.2), the degrees of vertices i G are roots of Eq. (2.2), so G must satisfy oe of the followig two cases: Case (A) G is regular; Case (B) G has oly two distict degrees.

4 Wag Yi, et al. ON GRAPHS WITH THREE DISTINCT LAPLACIAN EIGENVALUES 481 If case (A) holds, that is, G is d-regular (i.e. all vertices of G have the same degree d). By Eq. (2.3) ad Eq. (2.4) we fid that every pair of adjacet vertices has rs/ +2d (r + s) commo eighbors ad every pair of oadjacet vertices has rs/ commo eighbors. The G is said to be a strogly regular graph with parameters (, d, rs/ +2d (r + s),rs/)(see [14] Sectio 10.1]). Note that i this case the result of G beig strogly regular ca also be obtaied by Lemma of [14] from the fact L(G) =di A(G). If case (B) holds, that is, G has two distict degrees. By Eq. (2.4) every pair of distict oadjacet vertices has rs/ (rs/ 1) commo eighbors, the G has diameter 2 ad girth at most 5 if G is ot a tree. By the above discussio we have the followig result. Corollary 2.4. If G is a coected grapho vertices with three distict eigevalues 0,r,s(r s), the G either is regular or has two distict degrees. Furthermore, if G is d-regular the G is strogly regular with parameters (, d, rs/ +2d (r + s),rs/); if G has two distict degrees, the G has diameter 2 ad girth at most 5 if G is ot a tree. Lemma 2.5.(See [14], Lemma ) Let G be strogly regular with parameters (, d, a, c). Let the eigevalues of adjacecy matrix A(G) bed, adθ, τ respectively with multiplicities m θ,m τ. The either (a) G is a coferece graph (i.e. m θ = m τ ), ad hece d =( 1)/2, a =( 5)/4, c =( 1)/4; or (b) (θ τ) is a perfect square ad θ ad τ are itegers. Theorem 2.6. Let G be a regular graph with three distict eigevalues o vertices (ecessarily be a strogly regular graph). The G is Laplacia itegral if ad oly if G is ot a coferece graph or G is a coferece graph with beig a square of a odd umber. Proof. If G is ot a coferece graph, the by Lemma 2.5(b) ad the fact L(G) =di A(G), we get G is Laplacia itegral. If G is a coferece graph with parameters (, d, a, c), the by Lemma 2.5(a), d = ( 1)/2, a = ( 5)/4, c = ( 1)/4. By Corollary 2.4, a = rs/ +2d (r + s), c = rs/, soa c =2d (r + s) = 1 ad hece r + s =2d +1=, where r, s (r <s) are distict ozero eigevalues of L(G). We also have rs/ = c =( 1)/4, ad hece rs = ( 1)/4. So r = 2,s= + 2. As is odd, r, s are both itegers if ad oly if is a square of a odd umber. The result follows. Next we discuss the o-regular graphs with three distict eigevalues. Theorem 2.7. Let G be a o-regular ad triagle-free graph. The G has three distict eigevalues if ad oly if G is a star. I this case G is Laplacia itegral. Proof. By Corollary 2.4, if G has three distict eigevalues, the it has two distict degrees d 1,d 2 (d 1 >d 2 ), ad has diameter 2 ad girth at most 5 if G is ot a tree. Let the ozero eigevalues of G be r, s (r <s). Obviously, d 1,d 2 are two roots of Eq.(2.2), we have d 1 + d 2 = r + s 1. (2.6) By Eq. (2.3) ad the fact that G cotais o triagles, for each edge (v i,v j ) E(G), d vi + d vj = r + s rs/. (2.7) As G is coected, there exists a edge (v i,v j ) E(G) such that its vertices have respectively

5 482 Appl. Math. J. Chiese Uiv. Ser. B Vol. 22, No. 4 degrees of d 1 ad d 2. This implies that each edge of G jois two vertices with differet degrees d 1,d 2 ; otherwise it will coflict with Eq. (2.7). By this fact ad Eq. (2.6), we have rs/ =1. Let V 1,V 2 be the set of vertices of degrees d 1 ad d 2 respectively, the (V 1,V 2 )givesa bipartitio of the vertices of G. As each edge jois vertices with differet degrees, G is a bipartite graph. If there exists v i V 1,v j V 2 such that (v i,v j ) / E(G), by Eq. (2.4) we have N vi N vj = rs/ = 1, which implies that there exists a vertex v k of V 1 or V 2 such that v k is adjacet to both v i ad v j, ad hece there exists a edge withi V 1 or V 2, a cotradictio. Thus G is a complete bipartite graph. Also by Eq. (2.4), for ay two o-adjacet vertices v i,v j, N vi N vj =1,soG is a star ad the ecessity holds. The sufficiecy is easily verified. Oe may fid that a star o at least three vertices ad a complete regular bipartite graph o at least four vertices are graphs with three distict eigevalues. The above two graphs are both bipartite ad hece triagle-free. The latter property is a key assumptio i the proof of Theorem 2.7. Next we show that the above graphs are the oly oes amog bipartite graphs with three distict eigevalues. Note that a bipartite graph cotais o triagles, which is coveiet for our discussio i Theorem 2.6. Now we focus our problem o bipartite graphs. Lemma 2.8. [15] A r-regular graph G is bipartite graph if ad oly if λ i (G)+λ i (G) =2r for each i =1, 2,,. Theorem 2.9. A coected bipartite graph G has three distict eigevalues if ad oly if it is a complete regular bipartite graph or a star. I this case G is Laplacia itegral. Proof. Let G be a bipartite graph with two distict ozero eigevalues r, s (r s). By Corollary 2.4 we fid that either G is a regular bipartite graph or a bipartite graph with two distict degrees d 1,d 2. If G is a d-regular bipartite graph, by Lemma 2.8 we fid s =2d ad the multiplicity of s as a eigevalue of G is 1. Note that G is a bipartite graph ad hece cotais o triagles ad G cotais at least oe edge. By Eq.(2.3) we have rs/ +2d (r + s) = 0, ad hece rs/ = r ad s = as s =2d. So2d = ad G is a complete regular bipartite graph. If G is a bipartite graph with two distict degrees d 1,d 2, by Theorem 2.7, G is a star ad the ecessity holds. The sufficiecy is easily verified. For quadragle-free graph, from 4 i [10] we have the followig theorem. Theorem Let G be a o-regular ad quadragle-free graph. If G has two distict degrees d 1 ad d 2 (d 1 >d 2 ), X 1 ad X 2 deote the sets of vertices with degrees d 1 ad d 2,the G has three distict eigevalues if ad oly if X 2 iduces a coclique, maximal clique meetig both X 1 ad X 2 have size two, ad maximal cliques cotaied i X 1 have size d 1 d 2 +2, ad more, = d 1 d For petago-free graph G, before proceedig further alog this lie of discussio, let us pause a bit. If G is a graph of order, deote its complemet by G c. Observe that L(G) +L(G c )= I J,whereJ is matrix each of which etries is 1. It follows that λ i (G c )=λ i (G), 1 i<.

6 Wag Yi, et al. ON GRAPHS WITH THREE DISTINCT LAPLACIAN EIGENVALUES 483 If G 1 =(V 1,E 1 )adg 2 =(V 2,E 2 ) are graphs o disjoit sets of vertices, their uio is the graph G 1 + G 2 =(V 1 V 2,E 1 E 2 ). The joi, G 1 G 2, is the graph obtaied from G 1 + G 2 by addig ew edges from each vertex of G 1 to every vertex of G 2. Lemma Let G be a coected graph o vertices, the G has three distict eigevalues 0,r,,r,,,,adr, s with multiplicities p ad q if ad oly if (1) ( r 1) p, (2) q +1 p r 1, (3) G = K q+1 p G 1 G r 1 1 (Abbr. G r, ), where G 1 is r isolated vertices with p multiplicity r 1. Proof. By relatioship betwee eigevalues of G ad eigevalues of its complemet, result follows. Lemma [16] Let G 1 ad G 2 be graphs o disjoit sets of r ad s vertices, respectively. If S(G 1 )=(μ 1,,μ r )ads(g 2 )=(ν 1,,ν s ), the the eigevalues of G 1 G 2 are = r + s, μ 1 + s,,μ r 1 + s; ν 1 + r,,ν s 1 + r;0. Next we discuss the o-regular ad petago-free graphs with three distict eigevalues. Lemma Let G be a o-regular ad petago-free graph with vertices. The G has three distict eigevalues 0,r,,r,s,,sif ad oly if G has oe of the followig structures: (1) G = G r1, 1 K 2, (2) G = G r1, 1 G r2, 2,wherer = r 2 = 1. I this case G is Laplacia itegral. Proof. Let G be a o-regular ad petago-free graph with three distict eigevalues 0,r,,r,s,,s. By Eq.(2.4) i Corollary 2.3, ay two vertices that are ot adjacet have at least oe commo eighbor, so G does ot have a iduced subgraph isomorphic to P 4.By Theorem 2.5 i [11], G is decomposable(decomposable graphs are Laplacia itegral). That is, G ca be costructed from G 1 ad G 2 by uios, where G 1 with 1 vertices, G 2 with 2 vertices. By Lemma 2.12 the umber of distict eigevalues of G 1 (or G 2 )isomorethag. Hece, G 1 ad G 2 have at most three distict eigevalues. Evidetly, it is impossible that G 1 ad G 2 both have two distict eigevalues. So G 1 ad G 2 must satisfy oe of the followig two cases: Case(1) Oe of G 1 ad G 2 has two distict eigevalues, the other has three distict eigevalues. Case(2) G 1 ad G 2 both have three distict eigevalues. If case(1) holds, let G 1 ad G 2 have eigevalues 0,r 1,,r 1,s 1,,s 1 ad 0, 2,, 2, respectively. By Lemma 2.12, eigevalues of G 1 G 2 are 0,r 1 + 2,,r 1 + 2,s 1 + 2,,s 1 + 2, But G = G 1 G 2 oly have three distict eigevalues, hece s 1 = 1. That is, G 1 = G r1, 1. Result(1) holds. Similarly, we discuss case(2) ad get result(2). Sufficiecy is easily verified. Evidetly, i both cases G 1 ad G 2 are petago-free ad decomposable. If G 1 ad G 2 have three distict eigevalues, the they ca be operated by Theorem 2.13, too.

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