Expaer Graphs Graph Theory (Fall 011) Rutgers Uiversity Swastik Kopparty Throughout these otes G is a -regular graph 1 The Spectrum Let A G be the ajacecy matrix of G Let λ 1 λ λ be the eigevalues of A G Sometimes we will also be itereste i the Laplacia matrix of G This is efie to be L G = D A G, where D is the iagoal matrix where D vv equals the egree of the vertex v For -regular graphs, L G = I A G, a hece the eigevalues of L G are λ 1, λ,, λ Lemma 1 λ 1 = λ = λ 3 = = λ k = if a oly if G has at least k coecte compoets Proof For the first part, verify that the vector 1 is a eigevector with eigevalue For the seco part, let C 1,, C r be all the coecte compoets of G The 1 C1,, 1 Cr are all mutually orthogoal eigevectors with eigevalue if G has at least k coecte compoets, the λ = = λ r = Suppose there is a eigevector f ot i the spa of 1 C1,, 1 Cr with eigevalue Let C i be a compoet o which that eigevector is ot costat Let v C i be a vertex such that f(v) is maximum Sice f has eigevalue, f(v) = u Γ(v) f(u), a by maximality this implies that f(u) = f(v) for each u Γ(v) Repeatig this, we coclue that f must be costat o the compoet C i, a cotraictio I particular, coecte graphs have λ < We ow stuy the otio of expasio, where λ is less tha by a sigificat amout; this ca be thought of as a strog form of coecteess Expasio Defiitio (Eigevalue Expasio) We say a -regular graph is a λ eigevalue expaer if λ λ We say a -regular graph is a λ absolute eigevalue expaer if λ, λ λ If we have a family of graphs with teig to a with costat, we iformally call these graphs expaer graphs if they are all λ eigevalue expaers for some costat λ < Similarly we call these graphs absolute expaer graphs if they are all λ absolute eigevalue expaers for some costat λ < 1
Havig mae this efiitio, we ow show that families of expaer graphs (a absolute expaer graphs) exist Theorem 3 For every 3, there exists λ < such that for all sufficietly large, there exists a -vertex -regular λ absolute eigevalue expaer graph The proof is eferre to the e of these otes We first see why this otio is useful 3 Properties of Expaers May combiatorial properties of a graph ca be expresse i terms of the eigevalues of the ajacecy matrix I the case of eigevalue expaers, this coectio becomes very clea a powerful Let G = (V G, E G ) be a -regular -vertex graph with eigevalues = λ 1 λ λ Let v 1 = 1 1, v,, v be a orthoormal basis cosistig of eigevectors for λ 1,, λ respectively Let v : V G R The v ca be writte as a 1 v 1 + + a v, where a i = v, v i We have v = i=1 a i Note that v, L G v = x V G v(x) v, A G v = x V G y Γ(x) x V G y Γ(x) v(x)v(y), v(x)v(y) = 1 x V G y Γ(x) (v(x) v(y)) We ca also express these quatities i terms of the eigevalues We have A G v = a i λ i v i a L G v = a i ( λ i )v i Therefore v, A G v = a i λ i a v, L G v = a i ( λ i) Utilizig such expressios, we ca erive a large umber of combiatorial properties of expaer graphs Theorem 4 (Ege Expasio) Suppose G is a λ eigevalue expaer The for every S V G with S / we have e(s, S c ) λ S Proof We first wat to express the quatity e(s, S c ) i terms of the ajacecy matrix Let 1 S : V G R be the iicator fuctio of the set S The e(s, S c ) = 1 S, A G (1 1 S ) = S 1 S, A G 1 S Let S = i=1 a iv i We have a 1 = 1 S, 1 1 = S Further, we have a i = 1 S = S
So 1 S, A G 1 S = a i v i, a i λ i v i = a i λ i = S + a i λ i S S + λ( S ) = S ( λ) + λ S e(s, S c ) = ( λ) ) ( S S λ S Theorem 5 (Expaer Mixig Lemma) Suppose G is a λ absolute eigevalue expaer The for every S, T V G, we have e(s, T ) S T λ S T Proof Let 1 S = a k v i a 1 T = b i v i We have a 1 = S a a i = S, a similarly b 1 = T a b i = T The quatity of iterest e(s, T ) ca be expresse spectrally as follows: e(s, T ) = 1 S, A G 1 T = i a i v i, j b j λ j v j = i,j a i b j λ j v i, v j = i a i b i λ i = a 1 b 1 + = S T a i b i λ i + a i b i λ i The absolute value of seco term i this expressio ca be boue (usig the Cauchy-Schwarz iequality) by: ( ) 1/ ( ) 1/ λ b i λ S T This completes the proof a i 3
Theorem 6 (Vertex Expasio) Suppose G is a λ eigevalue expaer Let S V G The the eighborhoo of S is large: 1 Γ(S) S ( ) λ + 1 λ S I this theorem, if λ < (1 Ω(1)), a S < (1 Ω(1)), the Γ(S) = S (1 + Ω(1)), which is calle vertex expasio Proof Let 1 S = i a iv i The a 1 = S a a i = S Γ(S) is ot a quatity that ca be capture precisely i terms of the spectrum of G (ulike e(s, S c )) However it ca be lower boue usig the followig observatio: If f is a fuctio, the supp(f) f 1 f We apply this to f = A 1 S The supp(f) clearly equals Γ(S) We ow compute f 1 a f f 1 equals S f = A1 S = i a i λ i v i ( ( S + λ S 1 S )) 1/ Γ(S) S S + λ S S S = S + λ ( S ) 1 = S ( ) λ + 1 λ S This completes the proof Theorem 7 (Rapi Mixig of Raom Walks) Suppose G is a λ absolute eigevalue expaer Let v 0 V G be ay vertex Let µ be the probability istributio of the k-th step of the raom walk o G startig at v 0 The the statistical istace of µ from uiform is boue as follows: µ U 1 ( ) λ k, where U is the uiform istributio o V G Proof Let P be the matrix 1 A G If π R V G is a probability istributio o the vertices of G, the P π is the probability istributio of the vertex v, obtaie by pickig a vertex u accorig to π a the lettig v be a raom eighbor of that vertex Let π 0 be the probability istributio supporte o v 0 The probability istributio µ of the k-th step of the raom walk equals P k π 0 4
Let π 0 = i a iv i Note that a 1 = 1, a i a i = x V G π 0 (x) = 1 The µ = P k π 0 = i ( ) k λi a i v i = U + ( ) k λi a i v i Therefore µ U = ( a i ( ) λ k µ U 1 ( ) ) k 1/ λi ( ) λ k The ext theorem says that absolute eigevalue expaers have o large iepeet sets a hece they have large chromatic umber Theorem 8 (Iepeet Sets/Chromatic Number) The largest iepeet set i G has size at λ most λ the chromatic umber of G is at least λ λ You will prove this i your homework 4 Limits o eigevalue expasio How goo a eigevalue expaer ca a -regular graph be? This questio is aswere by a theorem of Alo a Boppaa Theorem 9 (Alo-Boppaa, simpler versio) Let G be a -regular graph which is a λ absolute eigevalue expaer The λ 1 o(1) (where the o(1) term tes to 0 as the umber of vertices tes to ifiity) 1 is ot just ay ol umber, it has very eep origis The ifiite ajacecy matrix of the ifiite -regular tree (this is the ultimate expaer graph) has its spectral raius equal to 1 Proof The mai iea is to stuy the eigevalues through the trace of a high eve power of A G We have: Tr(A k G ) = i=1 λ k i k + ( 1)λ k O the other ha, Tr(A k G ) couts the umber of close walks of legth k i G It is easy to see that the umber of such walks is at least as large as times the umber of such walks i the ifiite -regular tree The latter ca be coute by elemets σ of [] ([ 1] β) k 1 5
with exactly k β s, havig the property that ay prefix of σ of legth t has at most t/ β s This ca be coute easily i terms of Catala umbers; we get that the total umber of close walks i G of legth k is at least: ( 1 k + 1 Takig k = ω(1) a simplifyig, we get: ( ) k ) ( 1) k 1 k ( ( ) 1 k k + ( 1)λ k ) ( 1) k 1 k + 1 k λ 1 o(1) The full theorem of Alo-Boppaa shows that if G is a λ eigevalue expaer (ot ecessarily a absolute eigevalue expaer), eve the λ 1 o(1) This is a more elicate fact a the proof is more serious 5 Combiatorial versios of eigevalue expasio Defie the ege expasio of the graph G, eote h(g), by: h(g) = mi S V G, S / e(s, S c ) S h(g) beig large meas that most eges iciet o vertices of S are betwee S a S c By Theorem 4, we have that if G is a λ eigevalue expaer, the h(g) λ I particular, if λ = Ω(1), the h(g) Ω(1) We ow see the coverse of this statemet: if h(g) Ω(1), the G is a λ eigevalue expaer for λ Ω(1) Theorem 10 (Cheeger-Alo-Milma) Suppose λ R satisfies The G is a λ eigevalue expaer h(g) ( λ) Proof Let f : V G R be a uit orm eigevector of the eigevalue λ {x,y} E G (f(x) f(y)) = f, Lf = λ We will use f to fi a set S V G, with S /, such that e(s, S c ) ( λ ) S (this will show that h(g) ( λ ), as esire) First write f = f + f, where f + a f are oegative value Let V + a V be their supports Note that for each x V +, Lf + (x) Lf(x) = ( λ )f(x) = λ f + (x), 6
a for each x V, Lf (x) Lf(x) = ( λ )f(x) = ( λ )f (x) f +, Lf + ( λ ) f + a f, Lf ( λ ) f Now at least oe of the fuctios f + a f has support of size at most / Let us call that fuctio g We just showe that g, Lg ( λ ) g We will ow fi the esire S as a subset of the support of g It will be through a elicate probabilistic argumet Pick a [0, max x VG g(x)] uiformly at raom Let T = {x V G g(x) a} (ote that T is a raom set, with 0 < T /) The for a ege {x, y} E G, the probability that it gets coute i e(t, T c ) equals the probability that a lies i betwee g(x) a g(y), which equals g(x) g(y) : E[e(T, T c )] = g(x) g(y) {x,y} E G = (g(x) g(y)) (g(x) + g(y)) {x,y} E G 1/ (g(x) g(y)) (g(x) + g(y)) {x,y} E G {x,y} E G g, Lg 1/ ( g ( λ ) g ) 1/ O the other ha, E[ T ] = x V G g (x) = g E[e(T, T c )] E[ T ] ( λ ) This implies (verify this!) that with positive probability, e(t,t c ) T ( λ ), as esire Absolute eigevalue expasio ca also be characterize i terms of combiatorial properties of a graph A fuametal theorem of Bilu a Liial roughly says that a graph is a absolute eigevalue expaer if a oly if it satisfies the coclusio of the expaer mixig lemma 1/ 7