Appl. Math. J. Chiese Uiv. Ser. B 2007, 22(4): 478-484 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: 2006-08-28. MR Subject Classificatio: 05C50, 15A18. Keywords: Laplacia matrix, spectrum, Laplacia itegral, strogly regular graph. Digital Object Idetifier(DOI): 10.1007/s11766-007-0414-z. Supported by the Ahui Provicial Natural Sciece Foudatio (050460102), Natioal Natural Sciece Foudatio of Chia (10601001, 10571163), NSF of Departmet of Educatio of Ahui Provice (2004kj027, 2005kj005zd), Foudatio of Ahui Istitute of Architecture ad Idustry(200510307) ad Foudatio of Mathematics Iovatio Team of Ahui Uiversity, ad Foudatio of Talets Group Costructio of Ahui Uiversity.
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.
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.
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 10.2.1 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 10.3.3) 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
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 2.10. 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 2 +1. 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<.
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 2.11. 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 2.12. [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 2.13. 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 1 + 2 = 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, 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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