Linear combinations of graph eigenvalues

Transcription

1 Electroic Joural of Liear Algebra Volume 5 Volume Article Liear combiatios of graph eigevalues Vladimir ikiforov vikifrv@memphis.edu Follow this ad additioal works at: Recommeded Citatio ikiforov, Vladimir. 2006, "Liear combiatios of graph eigevalues", Electroic Joural of Liear Algebra, Volume 5. 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 LIEAR COMBIATIOS OF RAPH EIEVALUES VLADIMIR IKIFOROV Abstract. Let µ... µ be the eigevalues of the adjacecy matrix of a graph of order, ad be the complemet of. Suppose F is a fixed liear combiatio of µ i, µ i+,µ i, ad µ i+, i k. It is show that the limit lim max {F :v =} always exists. Moreover, the statemet remais true if the maximum is take over some restricted families like K r-free or r-partite graphs. It is also show that max µ +µ 2 2. v= 3 This iequality aswers i the egative a questio of erert. Key words. property. Extremal graph eigevalues, Liear combiatio of eigevalues, Multiplicative AMS subject classificatios. 5A, 05C50.. Itroductio. Our otatio is stadard e.g., see [], [3], ad [8]; i particular, all graphs are defied o the vertex set [] ={,...,} ad stads for the complemet of. We order the eigevalues ofthe adjacecy matrix ofa graph of order as µ... µ. Suppose k>0 is a fixed iteger ad α,...,α k,β,...,β k,γ,...,γ k,δ,...,δ k are fixed reals. For ay graph oforder at least k, let F = k α i µ i +β i µ i+ +γ i µ i + δi µ i+. i= For a give graph property F, i.e., a family of graphs closed uder isomorphism, it is atural to look for max {F : F, v =}. Questios ofthis type have bee studied; here is a partial list: max {µ +µ : is K r -free, v =} Bradt [2]; max {µ µ :v =} regory et al. [7]; max {µ +µ 2 :v =} erert [5]; max { } µ +µ : v = osal [], ikiforov [9]; max { } µ i +µ i : v = ikiforov [0]. Oe of the few sesible questios i such a geeral setup is the followig oe: does the limit lim max {F : F, v =} Received by the editors 7 August Accepted for publicatio 29 ovember Hadlig Editor: Stephe J. Kirklad. Departmet of Mathematical Scieces, Uiversity of Memphis, Memphis T 3852, USA, vkifrv@memphis.edu. 329

3 330 V. ikiforov exist? We show that, uder some mild coditios o F, this is always the case. For ay graph =V,E aditegert, write t for the graph obtaied by replacig each vertex u V by a set V u of t idepedet vertices ad joiig x V u to y V v ifad oly ifuv E. Call a graph property F multiplicative if: a F is closed uder addig isolated vertices; b Fimplies t Ffor every t. ote that K r -free, r-partite, ad ay graph are multiplicative properties. Theorem.. For ay multiplicative property F the limit. c = lim max {F : F, v =} exists. Moreover, { } c = lim sup F : F. ote that, sice the α i s, β i s, γ i s, ad δ i s may have ay sig, Theorem. implies that lim mi {F : F, v =} exists as well. erert [5] see also Stevaović [2] has proved that the iequality µ +µ 2 v holds ifthe graph has fewer tha 0 vertices or is oe of the followig types: regular, triagle-free, toroidal, or plaar; he cosequetly asked whether this iequality holds for ay graph. We aswer this questio i the egative by showig that < max µ +µ 2 2 <.55. v= 3 2. Proofs. ive a graph ad a iteger t>0, set [t] = t, i.e., [t] is obtaied from t by joiig all vertices withi V u for every u V. The followig two facts are derived by straightforward methods. i The eigevalues of t are tµ,...,tµ together with t additioal 0 s. ii The eigevalues of [t] are tµ +t,...,tµ +t together with t additioal s. We shall show that the extremal k eigevalues of t ad [t] are roughly proportioal to the correspodig eigevalues of.

4 Liear Combiatios of raph Eigevalues 33 Lemma 2.. Let k<,t 2. The for every s [k], 0 µ s t tµ s < t k, 0 µ s+ t tµ s+ > t, k 0 µ s [t] tµ s <t+ t, k 0 µ s+ [t] tµ s+ > t t. k Proof. We shall prove 2. first. Fix some s [k] ad ote that i implies that t ad have the same umber ofpositive eigevalues. I particular, t has at most egative eigevalues, ad so µ s t 0. If µ s t > 0, the µ s > 0adµ s t = tµ s, so 2. holds. If µ s t =0, the ad iequality 2. follows from 0 µ s... µ, k µ 2 s s +µ2 s µ 2 i <2. ext we shall prove 2.3. ote that ii implies that [t] ad have the same umber ofeigevalues that are greater tha. Sice [t] has at most eigevalues that are less tha, it follows that µ s [t]. If µ s [t] >, the µ s > adµ s [t] = tµ s +t ; thus, 2.3 holds. If µ s [t] =, the ad iequality 2.3 follows from i=s µ s... µ, k µ 2 s < s +µ 2 s µ 2 i < 2. Iequalities 2.2 ad 2.4 follow likewise with proper chages of sigs. We also eed the followig lemma. Lemma 2.2. Let be a graph of order ad H be a iduced subgraph of of order. The for every s 3/4, i=s µ s µ s H < 3, 0 µ s+ µ s H > 3.

5 332 V. ikiforov Proof. We shall assume that V ={,...,} ad V H ={,..., }. Let A be the adjacecy matrix of ad let A be the symmetric matrix obtaied from A by zeroig its th row ad colum. Sice the adjacecy matrix of H is the pricipal submatrix of A i the first colums ad rows, the eigevalues of A are µ H,...,µ H together with a additioal 0. This implies that, for every s [ ], { µs H, if µ 2.7 µ s A = s A > 0 µ s H if µ s A 0. We first show that, for every s [ ], 2.8 µ s A µ s H s. I view of2.7, this is obvious ifµ s A > 0. If µ s A 0, the we have Iequality 2.8 follows ow from µ s A µ s H =µ s H µ s H µ s H. s µ 2 s H s +µ 2 s H µ 2 i H < 2. Likewise, with proper chages ofsigs, we ca show that, for every s [ ], µ s+ A µ s H. s Havig proved 2.8, we tur to the proofof2.5 ad 2.6. ote that the first iequalities i both 2.5 ad 2.6 follow by Cauchy iterlacig theorem. O the other had, Weyl s iequalities imply that i=s µ A A µ s A µ s A µ A A. Obviously, µ A A is maximal whe the off-diagoal etries ofthe th row ad colum of A are s. Thus, µ A A adµ A A = µ A A. Hece, µ s µ s H =µ s A µ s A +µ s A µ s H + Likewise, s < 3. µ s+ µ s H =µ s+ A µ s+ A +µ s+ A µ s H > 3, s completig the proofoflemma 2.2.

6 Liear Combiatios of raph Eigevalues 333 Corollary 2.3. Let be a graph of order ad 2 be a iduced subgraph of of order l. The, for every s 3 l /4, µ s µ s 2 < 3l, µ s+ µ l s+ 2 < 3l. Proof. Let {v,...,v l } = V \V 2. Set H 0 = ; for every i [l], let H i be the subgraph of iduced by the set V \{v,...,v i } ; clearly, H l = 2. Sice H i+ is a iduced subgraph of H i with H i+ = H i, Lemma 2.2 implies that for every s 3 l /4, l l µ s µ s 2 µ s H i µ s H i+ 3 i<3l, i=0 l µ s+ µ l s+ 2 µ i+s+ H i µ i s+ H i+ i=0 l 3 i<3l, i=0 completig the proofofthe corollary. Proof of Theorem. Set ϕ = max {F : F, v =} Let M = k i= α i + β i + γ i + δ i ad set c = lim sup ϕ. Sice F M, the value c is defied. We shall prove that, i fact, c satisfies.. ote first if t 2, >4k/3, ad is a graph oforder, the for ay i [k], Lemma 2. implies that 2.9 F t tf M t + i=0 t k M t +2t 3Mt. Select ε>0adlet F be a graph oforder >3M/ε 2 such that Suppose max satisfies t max which implies the assertio. c + ε ϕ = F c ε. { 2, c /ε +, 3M/ε 2} ; therefore the value t = / { 2, c /ε +, 3M/ε 2}. We shall show that ϕ c 4ε,

7 334 V. ikiforov Let be the uio of t ad t isolated vertices. Clearly v = ad, sice F is multiplicative, F.Iviewof t <, Corollary 2.3 implies that F F t 3M. Therefore, i view of ϕ F / ad 2.9, ϕ F t 3M We fid that ϕ F ϕ F t ϕ tf 3Mt 3M. 3Mt +3M 3Mt +3M ϕ 2 c + ε +3Mt +3M ϕ 2 c + ε +3Mt t c + ε = ϕ t completig the proofoftheorem.. 3M 3M 3M t t c 4ε, We tur ow to the proofofiequality.2; we preset it i two propositios. Propositio 2.4. If is a graph of order, theµ +µ 2 2/ 3. Proof. Settig m = e, we see that 2.0 µ 2 +µ2 2 µ µ2 =2m. If m 2 /4, the result follows from µ +µ 2 2µ 2 +µ2 2 2 m, so we shall assume that m> 2 /4. From 2.0, we clearly have µ +µ 2 2m µ 2 2 +µ 2. The value 2m x 2 + x is icreasig i x for x m. O the other had, Weyl s iequalities imply that µ 2 +µ µ2 K =.

8 Liear Combiatios of raph Eigevalues 335 Hece, if K, we have µ 2 0adso,µ 2 2 <µ2 ;if = K, the µ 2 2 =µ2 + ; thus we always have µ 2 2 µ 2 +. From µ 2 2 µ 2 + e + + m m<m, we see that µ +µ 2 3m 2 /2+ 2 /2 m. The right-had side ofthis iequality is maximal for m =5 2 /2 ad the result follows. Propositio 2.5. For every 2 there exists a graph of order with µ +µ 2 > Proof. Suppose 2; set k = /2 ; let be the uio oftwo copies ofk 8k ad 2 be the joi of K 5k ad ; clearly v 2 =2k. Add 2k isolated vertices to 2 ad write for the resultig graph. By Cauchy s iterlacig theorem, we have µ µ 2, µ 2 µ 2 2 µ 2 =8k. Sice the graphs K 5k ad are regular, a theorem offick ad rohma [6] see also [3], Theorem 2.8 implies that µ 2 is the positive root ofthe equatio that is to say, x 5k +x 8k + 80k 2 =0, µ 2 = 3k 2+k Alteratively, applyig Theorem of[4], we see that Hece, µ 2 3k 2+k µ +µ 2 29k 4 + k > , >

9 336 V. ikiforov completig the proof. Ackowledgmet. Part ofthis research was completed while the author was visitig the Istitute for Mathematical Scieces, atioal Uiversity of Sigapore i The author is also idebted to Béla Bollobás for his kid support. Fially, the referees criticisms helped to correct the first versio ofthe paper. REFERECES [] B. Bollobás. Moder raph Theory. raduate Texts i Mathematics, 84, Spriger-Verlag, ew York, xiv+394 pp., 998. [2] S. Bradt. The local desity of triagle-free graphs. Discrete Math., 83:7 25, 998. [3] D. Cvetković, M. Doob, ad H. Sachs. Spectra of raphs. VEB Deutscher Verlag der Wisseschafte, Berli, 368 pp., 980. [4] C. odsil ad. Royle. Algebraic graph theory. raduate Texts i Mathematics, 207. Spriger- Verlag, ew York, xx+439 pp., 200. [5] D. erert, persoal commuicatio. [6] H.J. Fick ad. rohma. Vollstädiges Produkt, chromatische Zahl ud charakteristisches Polyom regulärer raphe. I. erma Wiss. Z. Tech. Hochsch. Ilmeau, : 3, 965. [7] D. regory, D. Hershkowitz, ad S. Kirklad. The spread of the spectrum of a graph. Liear Algebra Appl., 332/334:23 35, 200. [8] R. Hor ad C. Johso. Matrix Aalysis. Cambridge Uiversity Press, Cambridge, xiii+56 pp., 985. [9] V. ikiforov. Some iequalities for the largest eigevalue of a graph. Combi. Probab. Comput., :79 89, [0] V. ikiforov. Eigevalue problems of ordhaus-addum type. To appear i Discrete Math. [] E. osal. Eigevalues of raphs. Master s thesis, Uiversity of Calgary, 970. [2] D. Stevaović. arquivos/prob abertos.html.