Bounds for the Laplacian spectral radius of graphs Huiqing Liu a ; Mei Lu b a

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1 Ths artcle was ownloae by: [Tsnghua Unversty] On: 16 December 2009 Access etals: Access Detals: [subscrpton number ] Publsher Taylor & Francs Informa Lt Regstere n Englan an Wales Regstere Number: Regstere offce: Mortmer House, Mortmer Street, Lonon W1T 3JH, UK Lnear an Multlnear Algebra Publcaton etals, nclung nstructons for authors an subscrpton nformaton: Bouns for the Laplacan spectral raus of graphs Huqng Lu a ; Me Lu b a School of Mathematcs an Computer Scence, Hube Unversty, Wuhan, Chna b Department of Mathematcal Scences, Tsnghua Unversty, Bejng, Chna Frst publshe on: 02 January 2009 To cte ths Artcle Lu, Huqng an Lu, Me(2010) 'Bouns for the Laplacan spectral raus of graphs', Lnear an Multlnear Algebra, 58: 1, , Frst publshe on: 02 January 2009 (Frst) To lnk to ths Artcle: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms an contons of use: Ths artcle may be use for research, teachng an prvate stuy purposes. Any substantal or systematc reproucton, re-strbuton, re-sellng, loan or sub-lcensng, systematc supply or strbuton n any form to anyone s expressly forben. The publsher oes not gve any warranty express or mple or make any representaton that the contents wll be complete or accurate or up to ate. The accuracy of any nstructons, formulae an rug oses shoul be nepenently verfe wth prmary sources. The publsher shall not be lable for any loss, actons, clams, proceengs, eman or costs or amages whatsoever or howsoever cause arsng rectly or nrectly n connecton wth or arsng out of the use of ths materal.

2 Lnear an Multlnear Algebra Vol. 58, No. 1, January 2010, Bouns for the Laplacan spectral raus of graphs Huqng Lu a * an Me Lu b a School of Mathematcs an Computer Scence, Hube Unversty, Wuhan, Chna; b Department of Mathematcal Scences, Tsnghua Unversty, Bejng, Chna Communcate by B. Mohar (Receve 8 Aprl 2007; fnal verson receve 13 August 2008) Let G be a smple graph wth n vertces, m eges, ameter D an egree sequence 1, 2,..., n, an let 1 (G) be the largest Laplacan egenvalue of G. Denote ¼ max{ :1n}, ð tþ ¼ P j j an ð mþ ¼ð tþ =, where s a real number. In ths artcle, we frst gve an upper boun on 1 (G) for a non-regular graph nvolvng an D; next present two upper bouns on 1 (G) for a connecte graph n terms of an ( m) ; at last obtan a lower boun on 1 (G) for a connecte bpartte graph n terms of an ( t). Some known results are shown to be the consequences of our theorems. Keywors: graph; non-regular graph; Laplacan spectral raus AMS Subject Classfcatons: 05C50; 15A18 1. Introucton Let G ¼ (V, E) be a smple unrecte graph wth n vertces an m eges. Denote V(G) ¼ {v 1, v 2,..., v n }. For any two vertces v, v j 2 V(G), we wll use the symbol j to enote that vertces v an v j are ajacent. For v 2 V, the egree of v, wrtten by (v )or, s the number of eges ncent wth v. Let (G) ¼ an (G) ¼ be the mnmum egree an the maxmum egree of vertces of G, respectvely. The two egrees of v [2] s the sum of the egrees of the vertces ajacent to v an enote by t, an the average-egree of v s m ¼ t. Here we efne ð tþ ¼ X j j an ð mþ ¼ ð tþ, where s a real number. Note that ¼ ( 0 t) ¼ ( 0 m), t ¼ ( 1 t) an m ¼ ( 1 m). Let A(G) be the ajacency matrx of G an D(G) ¼ ag( 1, 2,..., n ) be the agonal matrx of vertex egrees. The Laplacan matrx of G s L(G) ¼ D(G) A(G). Clearly, L(G) s a real symmetrc matrx. From ths fact an Gersˇgorn s theorem, t follows that ts egenvalues are non-negatve real numbers. The egenvalues of an n n matrx M are enote by 1 (M), 2 (M),..., n (M), an we assume that 1 (M) 2 (M) n1 (M) n (M), whle for a graph G, we wll use (G) ¼ to enote (L(G)), ¼ 1, 2,..., n. Then 1 2 n1 n ¼ 0. We wll call (G), ¼ 1, 2,..., n, the Laplacan egenvalues an 1 (G) the Laplacan spectral raus of G. *Corresponng author. Emal: hql_2008@163.com ISSN prnt/issn onlne ß 2010 Taylor & Francs DOI: /

3 114 H. Lu an M. Lu The egenvalues of the Laplacan matrx L(G) can be use to obtan much nformaton about the graph, for example, estmates for expanng property, sopermetrc number, maxmum cut, nepenence number, genus, mean stance an ameter of a graph. In partcular, estmatng the boun for 1 (G) s of great nterest, an many results have been obtane [1,3 20]. In ths artcle, we gve some bouns of Laplacan spectral raus of G. From ths, we can mprove some known results, an hence these bouns are worthy of beng retane n terms of preceence (.e. for a gven set of graphs, how often oes the boun yel the best value among a gven set of bouns, see [3]). Let G be a connecte graph wth the egree agonal matrx D(G) an ajacency matrx A(G). Denote Q ¼ Q(G) ¼ D(G) þ A(G). Note that Q s non-negatve an rreucble. The Perron Frobenus Theorem mples that 1 (Q) s smple an has an egenvector x wth non-negatve entres whch must be postve f G s connecte. Now we state the followng lemma that wll be use n the proofs of our results. LEMMA 1.1 [19] Let G be a graph. Then 1 ðgþ 1 ðqþ: Moreover, f G s connecte, then the equalty hols f an only f G s a bpartte graph. 2. An upper boun for a non-regular graph It s well known that þ 1 1 ðgþ 2, where the rght-han equalty hols f an only f G s a -regular bpartte graph. [1,6] prove that f a connecte graph G s non-regular, then 2 1 ðgþ 5 2 nð2d þ 1Þ, ð2þ where D s the ameter of G. Motvate by [16] an [4], we show the followng stronger nequalty for a connecte non-regular graph G n ths secton. THEOREM 2.1 Let G be a smple connecte non-regular graph wth n vertces, m eges, ameter D an maxmum egree. Then 2n 4m 1 ðgþ 5 2 nðdð2n 4mÞþ1Þ : ð3þ Proof Let X ¼ (x 1, x 2,..., x n ) T be the unque unt postve egenvector of D þ A wth egenvalue 1 (Q). Then by Lemma 1.1, ð1þ ðqþ ¼2 Xn ð Þx 2 ðx x j Þ 2 : ð4þ ¼1 j,5j Choose vertces v s, v t so that x s ¼ max 1n {x } an x t ¼ mn 1n {x }. Snce G s non-regular, we replace P n ¼1 ð Þx 2 wth ðn 2mÞx 2 t n (4). Thus (3) hols by an argument smlar to the proof of Theorem 3.5 n [16]. g

4 Lnear an Multlnear Algebra 115 Note 2.2 Because the upper boun n (3) s monotone ecreasng n n 2m, t follows that nequalty (3) mproves the boun (2). 3. Two upper bouns for a connecte graph Throughout ths secton, let G be a smple graph of orer n wth egree sequence ( 1, 2,..., n ). Recall that ð tþ ¼ P j j an ð mþ ¼ð tþ =, where s a real number. Let ~D ¼ agf1,..., n g. THEOREM 3.1 Let G be a connecte graph. Then 8 q ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 9 < 2þ4ð þ j þ j mþ ð mþ j = 1 ðgþ mn max j : 2 ;, ð5þ where ranges over all real numbers. Proof By Lemma 1.1, 1 (G) 1 (Q). Moreover, 1 ðqþ ¼ 1 ð ~D 1 Q ~DÞ: Now, the (, j)th element of ~D 1 Q ~D s 8 f ¼ j, >< j f j, >: 0 otherwse: Let X ¼ (x 1, x 2,..., x n ) T be the egenvector corresponng to the egenvalue 1 ð ~D 1 Q ~DÞ. Let x ¼ max 1kn {x k }, an let x j ¼ max w {x w }. Snce ~D 1 Q ~DX ¼ 1 ð ~D 1 Q ~DÞX ¼ 1 ðqþx, we have Hence, from (6) an (7), we get ð 1 ðqþ Þx ¼ X w ð 1 ðqþ j Þx j ¼ X jw w w j x w ð mþ x j, x w ð mþ j x : ð6þ ð7þ Thus 2 1 ðqþð þ j Þ 1 ðqþþ j ð mþ ð mþ j 0: q 1 ðgþ ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 2þ4ð þ j þ j mþ ð mþ j : 2 Note 3.2 If ¼ 0, then the nequalty (5) s the Anerson an Morley s boun [1]. If ¼ 1, then the nequalty (5) s the Das boun [5, Theorem 2.14]. Let G be a graph shown n Fgure 1. Then the boun (5) s when ¼ 1.25, the Das boun s 6.5 an Anerson an Morley s boun s 8. Thus (5) s better than the Das boun an Anerson an Morley s boun. g

5 116 H. Lu an M. Lu Fgure 1. A graph G. From (6), we have the followng result. COROLLARY 3.3 Let G be a connecte graph. Then 1 ðgþ mn max þð mþ : ð8þ 1n Note 3.4 If ¼ 1, then the nequalty (8) s the Merrs boun [15]. The followng two lemmas wll be use n the proof of next theorem. LEMMA 3.5 [12] Let B be the ajacency matrx of the lne graph of G. If s the largest egenvalue of B þ 2I, where I s the entty matrx, then 1. LEMMA 3.6 [2] Let A be an n n matrx wth spectral raus (A) an x be an n-tuple postve vector. Then ðaþ max ðaxþ : 1 n, x where x s the th component of x. Now, we gve another man result of ths secton. THEOREM 3.7 Let G be a connecte graph. Then ( 1 mn max ð þð mþ Þþj ð ) j þð mþ j Þ j þ j : Proof Let B be the ajacency matrx of the lne graph of G. If j, x y, then B(v v j, v x v y ) ¼ 1fv v j an v x v y are ajacent, an 0, otherwse. Let W be a column vector whose j th component s þ j. Then the j th component of (B þ 2I)W s 2 þ j x þ y ¼ 2 þ j jxy jy þ y x þ j jxj

6 ¼ 2 þ j y Lnear an Multlnear Algebra 117 þ y þ j jx x þ j j þ Thus ¼ þ1 þ ð mþ þ j þ1 þ j ð mþ j ¼ ð þð mþ Þþ j ð j þð mþ j Þ: ððb þ 2IÞWÞ j ¼ W j þð mþ þ j j þð mþ j þ j : The result hols from Lemmas 3.5 an 3.6. g Note 3.8 If ¼ 1, then our result n Theorem 3.7 s the L an Zhang s boun [12, Theorem 3]. In [3], Brankov et al. prove automate ways to generate two large sets of conjecture upper bouns on the largest Laplacan egenvalue of graphs. In one set, they consere a smlar form of (conjecture) upper boun, epenng on the eges of G, that s: 1 max f ð, m, j, m j Þ: j If we replace the bounng functon f (, m, j, m j ) by f (,( m), j,( m) j ) or by f ð, ð mþ, j, ð mþ j Þ an follow the generatng steps, we shoul get the bouns of Theorems 3.1 an 3.7, respectvely. We beleve that smlar results can be obtane for bouns of those types by the automate ways n [3]. An we are sure that the technques of provng Theorems 3.1 an 3.7 are able to prove a substantal number of such bouns. 4. A lower boun for a bpartte graph In ths secton, we gve a lower boun for the largest egenvalue of the Laplacan matrx of a bpartte graph. Recall that ð tþ ¼ P j j. Frst we state the followng lemma. LEMMA 4.1 [8] Let A be a non-negatve symmetrc matrx an X be a unt vector of R n. If (G) ¼ X T AX, then AX ¼ (A)X. THEOREM 4.2 Let G be a connecte bpartte graph of orer n wth egree sequence 1,..., n. Then 8vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff9 >< up n ¼1 1 ðgþ max þ1 þð 2 tþ >= t P n 2< >: ¼1 2 >; : ð9þ Moreover, the equalty hols n (9) for a partcular value of 51 f an only f G s a regular graph. Proof Note that Q ¼ D þ A an L ¼ D A have the same non-zero egenvalues by G beng a bpartte graph an Q s a non-negatve rreucble postve semefnte

7 118 H. Lu an M. Lu symmetrc matrx. Let X ¼ (x 1, x 2,..., x n ) T be the unt postve egenvector of Q corresponng to 1 (Q). Take sffffffffffffffffffffffffffffffffffff 1 C ¼ P n ¼1 2 1, 2,..., n T: Then, by an argument smlar to the proof of Theorem 3.3 n [16], we have vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff up n ¼1 1 ðgþ þ1 þð 2 t tþ P n ¼1 2 an the equalty n (9) hols for a partcular value of only f C s an egenvector of 1 (Q), whch mples that for all 1 n, þ1 þð tþ ¼ 1 ðqþ,.e. þð mþ ¼ 1 : Suppose v, v j 2 V(G) wth ¼ an j ¼. Then for 40, we have an Thus, for 40, we have Smlarly, for 0, 1 ¼ þð mþ j þ 1 ¼ð þ Þ 1 : 1 ¼ þð mþ þ 1 ¼ð þ Þ 1 : ð þ Þ 1 1 ð þ Þ 1 : 2 ¼ þ 1 1 þ 1 ¼ 2: Thus, combnng these two cases, we have ¼ for 51. So G s regular. Conversely, f G s -regular, then þ1 þð tþ ¼ 2 þ1. It follows that vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff up n ¼1 1 ðgþ ¼2 ¼ þ1 þð 2 t tþ P n ¼1 2 : Note 4.3 If ¼ 1 2, then the nequalty (9) s the Sh s boun [16, Theorem 3.3]; f ¼ 1, then the nequalty (9) s the boun by Yu et al. [18, Theorem 9]; f ¼ 0, then the nequalty (9) s the Hong s boun (also see Corollary 10 of [18]). g Acknowlegements Many thanks to the anonymous referee for hs/her many helpful comments an suggestons, whch have conserably mprove the presentaton of the artcle. H. Lu was partally supporte by NNSFC (Nos , ) an Scentfc Research Fun of Hube Provncal Eucaton Department (No. D ). M. Lu was partally supporte by NNSFC (No ).

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