Characterizing graphs of maximum principal ratio

Transcription

1 Characterizig graphs of maximum pricipal ratio Michael Tait ad Josh Tobi November 9, 05 Abstract The pricipal ratio of a coected graph, deoted γg, is the ratio of the maximum ad miimum etries of its first eigevector Cioabă ad Gregory cojectured that the graph o vertices maximizig γg is a kite graph: a complete graph with a pedat path I this paper we prove their cojecture Itroductio Several measures of graph irregularity have bee proposed to evaluate how far a graph is from beig regular I this paper we determie the extremal graphs with respect to oe such irregularity measure, aswerig a cojecture of Cioabă ad Gregory [5] All graphs i this paper will be simple ad udirected, ad all eigevalues are of the adjacecy matrix of the graph For a coected graph G, the eigevector correspodig to its largest eigevalue, the pricipal eigevector, ca be take to have all positive etries If x is this eigevector, let x mi ad x max be the smallest ad largest eigevector etries respectively The defie the pricipal ratio, γg to be γg = x max x mi Note that γg with equality exactly whe G is regular, ad it therefore ca be cosidered as a measure of graph irregularity Let P r K s be the graph attaied by idetifyig a ed vertex of a path o r vertices to ay vertex of a complete graph o s vertices This has bee called a kite graph or a lollipop graph Cioabă ad Gregory [5] cojectured that the coected graph o vertices maximizig γ is a kite graph Our mai theorem proves this cojecture for large eough Theorem For sufficietly large, the coected graph G o vertices with largest pricipal ratio is a kite graph We ote that Brightwell ad Wikler [4] showed that a kite graph maximizes the expected hittig time of a radom walk Other irregularity measures for graphs have bee well studied Bell [3] studied the irregularity measure ɛg := λ G dg, the differece betwee the spectral radius ad the average degree of G He determied the extremal graph over all ot ecessarily coected graphs o vertices ad e edges It is ot kow what the extremal coected graph is, ad Aouchiche et al [] cojectured that this extremal graph is a pieapple : Both authors were partially supported by NSF grat DMS {mtait, rjtobi}@mathucsdedu

2 a complete graph with pedat vertices added to a sigle vertex Bell also studied the variace of a graph, varg = dv d v V G Albertso [] defied a measure of irregularity by du dv uv EG ad the extremal graphs were characterized by Hase ad Mélot [6] Nikiforov [9] proved several iequalities comparig varg, ɛg ad sg := v du d Bell showed that ɛg ad varg are icomparable i geeral [3] Fially, bouds o γg have bee give i [5, 0, 8, 7, ] Prelimiaries Throughout this paper G will be a coected simple graph o vertices The eigevectors ad eigevalues of G are those of the adjacecy matrix A of G The vector v will be the eigevector correspodig to the largest eigevalue λ, ad we take v to be scaled so that its largest etry is Let x ad x k be the vertices with smallest ad largest eigevector etries respectively, ad if several such vertices exist the we pick ay of them arbitrarily Let x, x,, x k be a shortest path betwee x ad x k Let γg be the pricipal ratio of G We will abuse otatio so that for ay vertex x, the symbol x will refer also to vx, the value of the eigevector etry of x For example, with this otatio the eigevector equatio becomes λv = w v w We will make use of the Rayleigh quotiet characterizatio of the largest eigevalue of a graph, λ G = max 0 v v T AGv v t v Recall that the vertices v, v,, v m are a pedat path if the iduced graph o these vertices is a path ad furthermore if, i G, v has degree ad the vertices v,, v m have degree ote there is o requiremet o the degree of v m Lemma If λ ad σ = λ + λ 4/, the for j k, γg σj σ j x σ σ j Moreover we have equality if the vertices x, x,, x j are a pedat path Proof We have the followig system of iequalities λ x x λ x x + x 3 λ x 3 x + x 4 λ x j x j + x j

3 The first iequality implies that x λ x Pluggig this ito the secod equatio ad rearragig gives Now assume that with u j positive for all j < i The x λ λ x 3 x i u i u i x i+ λ x i+ x i + x i+ implies that x i+ u i λ u i u i x i+ where λ u i u i must be positive because x j is positive for all j Therefore the coefficiets u i satisfy the recurrece u i+ = λ u i u i Solvig this ad usig the iitial coditios u 0 =, u = λ we get u i = σi+ σ i σ σ I particular, u i is always positive, a fact implicitly used above Fially this gives, Hece x u 0 u x u 0 u u u x 3 x j u j γg = x k = σj σ j x x x σ σ j If these vertices are a pedat path, the we have equality throughout We will also use the followig lemma which comes from the paper of Cioabă ad Gregory [5] Lemma 3 For r ad s 3, s + ss < λ P r K s < s + s I the remaider of the paper we prove Theorem We ow give a sketch of the proof that is cotaied i Sectio 3 We show that the vertices x, x,, x k are a pedat path ad that x k is coected to all of the vertices i G that are ot o this path lemma 5 Next we prove that the legth of the path is approximately / log lemma 6 3 We show that x k has degree exactly lemma 9, which exteds our pedat path to x, x,, x k To do this, we fid coditios uder which addig or deletig edges icreases the pricipal ratio lemma 7 4 Next we show that x k also has degree exactly lemma At this poit we ca deduce that our extremal graph is either a kite graph or a graph obtaied from a kite graph by removig some edges from the clique We show that addig i ay missig edges will icrease the pricipal ratio, ad hece the extremal graph is exactly a kite graph 3

4 3 Proof of Theorem Let G be the graph with maximal pricipal ratio amog all coected graphs o vertices, ad let k be the umber of vertices i a shortest path betwee the vertices with smallest ad largest eigevalue etries As above, let x,, x k be the vertices of the shortest path, where γg = x k /x Let C be the set of vertices ot o this shortest path, so C = k Note that there is o graph with k =, as the edpoits of a path have the same pricipal eigevector etry Also λ G, otherwise P K 3 would have larger pricipal ratio Fially ote that k is strictly larger tha, otherwise x k = x ad G would be regular Lemma 4 λ G > k Proof Let H be the graph P k K k+ It is straightforward to see that i H, the smallest etry of the pricipal eigevector is the vertex of degree ad the largest is the vertex of degree k + Also ote that i H, the vertices o the path P k form a pedat path By maximality we kow that γg γh Combiig this with lemma, we get where σ H = Now the fuctio σ k σ k σ σ λ H + λ H 4 / γg γh = σk H σ k H σ H σ H fx = xk x k x x is icreasig whe x Hece we have σ σ H, ad so λ G λ H > k Lemma 5 x, x,, x k are a pedat path i G, ad x k is coected to every vertex i G that is ot o this path Proof By our choice of scalig, x k = From lemma 4 k < λ G = y x k y Nx k Now Nx k is a iteger, so we have Nx k k+ Moreover because x, x,, x k is a iduced path, we must have that Nx k = k+ exactly, ad hece the Nx k = C {x k } It follows that x, x,, x k 3 have o eighbors off the path, as otherwise there would be a shorter path betwee x ad x k Lemma 6 For the extremal graph G, we have k = + o Proof Let H be the graph P j K j+ where j = log, ad let G be the coected graph o vertices with maximum pricipal ratio Let x,, x k be a shortest path from x to x k where γg = x k x By lemma 5, we have By the eigevector equatio, this gives that λ G G k + log γg k + k Now, lemma gives that γh = σj H σ j H σ H σ, H 4

5 where σh = λ H + λ H 4 Now, s + ss < λ P r K s < s + s, so we may choose large eough that log + > σ H σ H > log By maximality of γg, we have Thus, k = + o log k + k γg γh log log For the remaider of this paper we will explore the structure of G by showig that if certai edges are missig, addig them would icrease the pricipal ratio, ad so by maximality these edges must already be preset i G We have established that the vertices x, x,, x k are a pedat path, ad so we have γg = σk σ k+ σ σ x k 3 We will ot add ay edges that affect this path, ad so the above equality will remai true The chage i γ is the completely determied by the chage i λ ad the chage i x k The ext lemma gives coditios o these two parameters uder which γ will icrease or decrease Lemma 7 Let x, x,, x m form a pedat path i G, where m = + o/ log Let G + be a graph obtaied from G by addig some edges from x m to V G \ {x,, x m }, where the additio of these edges does ot affect which vertex has largest pricipal eigevector etry Let λ + be the largest eigevalue of G + with leadig eigevector etry for vertex x deoted x +, also ormalized to have maximum etry oe Defie δ ad δ such that λ + = + δ λ ad x + m = + δ x m The γg + > γg wheever δ > 4δ / γg + < γg wheever δ expδ λ log < δ /3 Proof We have So σ = λ λ λ 3 λ 5 3 λ + λ < σ + σ < λ + λ λ + λ λ whe λ is sufficietly large, which is guarateed by lemma 6 Pluggig i λ + = + δ λ, we get δ λ < σ + σ < δ λ + λ + δ < δ λ + δ I particular + δ /σ < σ + < + δ σ To prove part i, we wish to fid a lower boud i the chage i the first factor of equatio 3 Let fx = xm x m+ x x 5

6 The mx m 3 > f x > m x m 3 mx m 5, ad usig that m / log ad σ λ which goes to ifiity with, we get f x m x m 3 By liearizatio ad because fσ σ m, it follows that σ+ m σ+ m+ σ + σ+ + δ m 3 σ m σ m+ σ σ Hece, if δ m 3 > δ the γg + > γg I particular it is sufficiet that δ > 4δ / To prove part ii, recall from above that f x < mx m 3 The, whe x = + o/ log So for 0 < ε < δ λ, we have f x + ε < mx + ε m 3 m 3 = mx m 3 + ε x mε mx m 3 exp x x m 3 exp logε f x + ε < x m 3 exp logδ λ Hece + 3 expδ λ log δ σ m σ m+ σ σ > σm + σ+ m+ σ + σ+ Lemma 8 For every subset of U of Nx k, we have U < y U y U A immediate cosequece is that there is at most oe vertex i the eighborhood of x k with eigevector etry smaller tha / Proof The upper boud follows from y, ad the lower boud from the iequalities y Nx k U y Nx k \U ad y Nx k y = λ G > Nx k Lemma 9 The vertex x k has degree exactly i G 6

7 Proof Assume to the cotrary Let U = Nx k Nx k The U, so by lemma 8 we have y > U Now, by the same argumet as the i the proof of lemma, we have that y U γg = σk σ k+ σ σ y y U Let H = P k K k+ The by maximality of γg we have σ k σ k+ σ σ > γg γh = σk H σ k+ H σ H σ H So σ > σ H, which meas λ G > λ H > k + This meas that G > k +, but we have established that G = k + We ow kow that x, x,, x k is a pedat path i G, ad so equatio 3 becomes γg = σk σ k+ σ σ x k 4 Lemma 0 The vertex x k has degree less tha C / log Proof Assume to the cotrary, so throughout this proof we assume that the degree of x k is at least C / log Let G + the graph obtaied form G with a additioal edge from x k to a vertex z C with z / Let λ + = λ G + ad let x + be the pricipal eigevector etry of vertex x i G +, where this eigevector is ormalized to have x + k = Chage i λ : By equatio, we have λ + λ x k z A crude upper boud o v v is We also have that z / so v + y λ y x λ + < λ k λ + + x k λ λ Chage i x k : Let U = Nx k C By the eigevector equatio we have x k = x k + x k + y λ y U x + k = λ + x + k + x+ k + z+ + y + y U Subtractig these, ad usig that λ < λ + ad x k = x + k =, we get x + k x k x + k λ x k + z + + y + y y U 7

8 By lemma 8, we have y U y+ y We also have x + k x k < ad z + Hece x + k x k 3/λ, or x + k 3 + x k λ x k We ca oly apply lemma 7 if x + k is the largest eigevector etry i G + So we must cosider two cases Case : If i G + the largest eigevector etry is still attaied by vertex x k, the we ca apply lemma 7, ad see that γg + > γg if or equivaletly x k λ λ x k x k 4λ We have that λ = + o / log, so it suffices for We kow that By assumptio x k 5 log 5 x k > U λ U + = Nx k C / log Equatio 5 follows from this, so γg + > γg Case : Say the largest eigevector etry of G + is o loger attaied by vertex x k It is easy to see that the largest eigevector etry is ot attaied by a vertex with degree less tha or equal to, ad comparig the eighborhood of ay vertex i C with the eighborhood of x k we ca see that x k y for all y C So the largest eigevector etry must be attaied by x k The equatio 4 o loger holds, istead we have γg + = σk + σ+ k+ σ + σ 6 Recall that i lemma 7 we determied the chage from γg + to γg by cosiderig λ + λ ad x + k x k I this case, by 6, we must cosider λ + λ ad x k Now if x + k > x+ k, the vertex x k i G is coected to all of C except perhaps a sigle vertex Hece i G, the vertex x k is coected to all of C except at most two vertices This gives the boud + x k 3/λ ad so as i the previous case, γg + > γg So i all cases, x k is coected to all vertices i C that have eigevector etry larger tha / If all vertices i C have eigevector etry larger tha /, the x k is coected to all of C, ad this implies that x k > x k, which is a cotradictio At most oe vertex i C is smaller tha /, ad so there is a sigle vertex z C with z < / We will quickly check that addig the edge {x k, z} icreases the pricipal ratio As before let G + be the graph obtaied by addig this edge The largest eigevector etry i G + is attaied by x k, as its eighborhood strictly cotais the eighborhood of x k As above, addig the edge {z, x k } icreases the spectral radius at least λ + > + z λ λ 8

9 ad we have x k < z/λ Applyig lemma 7 we see that γg + > γg, which is a cotradictio Fially we coclude that the degree of x k must be smaller tha C / log We ote that this lemma gives that x k < / which implies that ay vertex i C has eigevector etry larger tha / Lemma The vertex x k has degree exactly i G It follows that x k < /λ Proof Let U = Nx k C, c = U If c = 0 the we are doe Otherwise let G be the graph obtaied from G by deletig these C edges We will show that γg > γg Chage i λ : We have by equatio, λ λ cx k v By Cauchy Schwarz, v > x x Nxk x C + x Nx k k k + We also have Combiig these we get We have λ λ x k c + λ λ λ < 9c λ k + λ < > k, so λ < + + 0c k 3 λ 9c λ λ k + Chage i x k : At this poit, we kow that i G the vertices x,, x k form a pedat path, ad so by the proof of lemma, we have x k = +o/λ By the eigevector equatio ad usig that the vertices i C have eigevector etry at least /, we have x k > +c//λ So x k x k > λ c + o I particular, c x k > + 3x k λ Applyig lemma 7, it suffices ow to show that 0c k 3 exp 0c k 3 λ log x k < λ c 9x k λ 7 Now 0c < 0 k 3 log C k 3 < 3 log log = 3 9

10 Similarly 0c k λ 3 log < 3, so the lefthad side of equatio 7 is smaller tha C 0 /, where C 0 is a absolute costat For the righthad side, recall that x k λ = + o, ad also that c > log log + o > 0 log 3/ So the righthad side is larger tha / log 3/ Hece for large eough, the righthad side is larger tha the lefthad side We are ow ready to prove the mai theorem Theorem For sufficietly large, the coected graph G o vertices with largest pricipal ratio is a kite graph Proof It remais to show that C iduces a clique Assume it does ot, ad let H be the graph P k K k+ We will show that γh > γg, ad this cotradictio tells us that C is a clique As before, lemma gives that where γh = σk H σ k H σ H σ, H σh = λ H λ H 4 Sice x, x k form a pedat path we also kow that γg = σk σ k σ σ Now, λ H > λ G because EG EH Sice the fuctios gx = x + x 4 ad fx = x k x k /x x are icreasig whe x, we have γh > γg Ackowledgemets We would like to thak greatly Xig Peg for helpful discussios ad commets o a earlier draft of this paper Refereces [] Michael O Albertso The irregularity of a graph Ars Combiatoria, 46:9 5, 997 [] Mustapha Aouchiche, Fracis K Bell, D Cvetković, Pierre Hase, Peter Rowliso, SK Simić, ad D Stevaović Variable eighborhood search for extremal graphs 6 some cojectures related to the largest eigevalue of a graph Europea Joural of Operatioal Research, 93:66 676, 008 [3] Fracis K Bell A ote o the irregularity of graphs Liear Algebra ad its Applicatios, 6:45 54, 99 [4] Graham Brightwell ad Peter Wikler Maximum hittig time for radom walks o graphs Radom Structures & Algorithms, 3:63 76, 990 [5] Sebastia M Cioabă ad David A Gregory Pricipal eigevectors of irregular graphs Electro J Liear Algebra, 6: , 007 0

11 [6] Pierre Hase, Hadrie Mélot, ad Québec Groupe d études et de recherche e aalyse des décisios Motréal Variable eighborhood search for extremal graphs 9: Boudig the irregularity of a graph Motréal: Groupe d études et de recherche e aalyse des décisios, 00 [7] Geoff A Latham A remark o mic s maximal eigevector boud for positive matrices SIAM Joural o Matrix Aalysis ad Applicatios, 6:307 3, 995 [8] Heryk Mic O the maximal eigevector of a positive matrix SIAM Joural o Numerical Aalysis, 73:44 47, 970 [9] Vladimir Nikiforov Eigevalues ad degree deviatio i graphs Liear Algebra ad its Applicatios, 44: , 006 [0] Britta Papedieck ad Peter Recht O maximal etries i the pricipal eigevector of graphs Liear Algebra ad its Applicatios, 30:9 38, 000 [] Xiao-Dog Zhag Eigevectors ad eigevalues of o-regular graphs Liear Algebra ad its Applicatios, 409:79 86, 005