Spectral Graph Theory ad Applicatios WS 20/202 Lecture 2: Spectra of Graphs Lecturer: Thomas Sauerwald & He Su Our goal is to use the properties of the adjacecy/laplacia matrix of graphs to first uderstad the structure of the graph ad, based o these isights, to desig efficiet algorithms. The study of algebraic properties of graphs is called algebraic graph theory. Oe of the most useful algebraic properties of graphs are the eigevalues (ad eigevectors) of the adjacecy/laplacia matrix. Defiitios Defiitio 2.. Let G = (V, E) be a udirected graph with vertex set [] := {,..., }. The adjacecy matrix of G is a by matrix A give by { if i ad j are adjacet A i,j = 0 otherwise If G is a multi-graph, the A i,j is the umber of edges betwee vertex i ad vertex j. The sum of elemets i every row/colum equals the degree of the correspodig vertex. If G is udirected, the A is symmetric. Example. The adjacecy matrix of a triagle is 0 0 0 Defiitio 2.2. We defie the Laplacia matrix L of graph G as follows: deg(i) if i = j L i,j = if i ad j are adjacet 0 otherwise where deg(i) is the degree of vertex i. Let D be a by diagoal matrix with deg(),..., deg() as diagoal elemets. We ca rewrite L as L = D A. I particular, if G is d-regular, the L = d I A. Give a matrix A, a vector x 0 is defied to be a eigevector of A if ad oly if there is a λ C such that Ax = λx. I this case, λ is called a eigevalue of A.
Defiitio 2.3 (graph spectrum). Let A be the adjacecy matrix of a udirected graph G with vertices. The A has real eigevalues, deoted by λ λ. These eigevalues associated with their multiplicities compose the spectrum of G. Here are some basic facts about the graph spectrum. Lemma 2.4. Let G be ay udirected simple graph with vertices. The. i= λ i = 0. 2. i= λ2 i = i= deg(i). 3. If λ = = λ, the E[G] =. 4. deg avg λ deg max. 5. deg max λ deg max. Proof. We oly prove the first three items. () Sice G does ot have self-loops, all the diagoal elemets of A are zero. By the defiitio of trace, we have ı= λ i = tr(a) = i= A i,i = 0. (2) By the properties of matrix trace, we have ı= λ2 i = tr ( A 2) = i= A2 i,i. Sice A2 i,i is the degree of vertex i, tr(a 2 ) equals the sum of every vertex s degree i G. (3) Combig i= λ i = 0 with λ = = λ, we have λ i = 0 for every vertex i. By item (2), we have deg(i) = 0 for ay vertex i. Therefore E[G] =. For a graph G with adjacecy matrix A ad iteger k, A k u,v is the umber of walks of legth k from u to v. Let x p := ( i= x i p ) /p. The for ay p q <, it holds that x q x p /p /q x q. Lemma 2.5. For ay graph G with m edges, the umber of cycles of legth k i G is bouded by O ( m k/2). Proof. Let A be the adjacecy matrix of G with eigevalues λ,..., λ. Thus the umber of C k (cycles of legth k) i G is bouded by tr ( A k) /(2k) = ( ) i= λk i /(2k). For ay k 3, it holds that ( ) /k ( ) /k ( ) /2 λ k i λ i k λ i 2 = (2 m) /2. i= i= Hece tr ( A k) (2m) k/2 ad the umber of C k is at most O ( m k/2). Example. Some examples for differet spectra of graphs: For the complete graph K, the eigevalues are with multiplicity ad with multiplicity. For the complete bipartite graph K m,, the eigevalues are + m, m ad 0 with multiplicity m + 2. For the cycle C, the spectrum is 2 cos(2πj/) (j = 0,,..., ). Two assumptios that we make throughout the course are as follows: i= 2
. We oly cosider udirected graphs. Note that if G is ot udirected, the A ad L is ot symmetric ay more ad the eigevalues of A ad L could be complex umbers. A matrix A = (a i,j ) is called a Hermitia matrix if a i,j = a j,i for ay elemet a i,j. Hermitia matrices always have real eigevalues. 2. Uless metioed otherwise, we cosider regular graphs. Lemma 2.6. Cosider ay udirected graph G with adjacecy matrix A.. If G is d-regular, the λ = d ad λ i d for i = 2,...,. 2. G is coected iff λ 2 < d, i.e., the eigevalue d has multiplicity. Moreover, the umber of coected compoets of G equals the multiplicity of eigevalue d. 3. If G is coected, the G is bipartite iff λ = d. Lemma 2.7. All eigevalues of L are o-egative. Proof. Follows from Lemma 2.6 ad the defiitio of L = D A. For studyig regular graphs, it is coveiet to work with the ormalized adjacecy matrix M of graph G. For ay d-regular graph with adjacecy matrix A, defie M := d A. Throughout this course, we use λ λ to deote the eigevalues of matrix M of graph G. For regular graphs, λ = ad we maily cosider the secod largest eigevalue i absolute value. The formal defiitio is as follows. Defiitio 2.8 (spectral expasio). The spectral expasio of graph G is defied by λ := max { λ 2, λ }, i.e. λ = max Ax x =,x u Courat-Fischer Formula. Let B be a by symmetric matrix with eigevalues λ λ ad correspodig eigevectors v,..., v. The x T Bx λ = mi x = xt Bx = mi x 0 x T x, λ 2 = mi x = x v x T x T Bx Bx = mi x 0 x T x, x v x T Bx λ = max x = xt Bx = max x 0 x T x. It is well kow that λ relates to various graph properties. I particular, we shall see that there is a close coectio betwee λ ad the expasio of the graph. Lemma 2.9. d λ d( ). 3
Proof. Follows from tr ( M 2) = /d = i= λ2 i + ( )λ2. Theorem 2.0. [Alo86] Ay ifiite family of d-regular graphs {G } N has spectral expasio (as ) at least 2 d /d o(). Defiitio 2. (Ramauja graphs). A family of d-regular graphs with spectral expasio at most 2 d /d is called Ramauja graphs. Although Friedma [Fri9] showed that radom d-regular graphs are close to beig Ramauja i the sese that λ satisfies λ 2 d /d + 2 log(d)/d + o(), costructig families of Ramauja graphs with arbitrary degree is oe of the biggest ope problems i this area. So far, we oly kow the costructio of Ramauja graphs with certai degrees ad these costructios are based o deep algebraic kowledge. See [LPS88] for example. Aother quite importat problem is to fid a combiatorial costructio of Ramauja graphs. At the ed of this sectio, we list some more iterestig facts o eigevalues of graphs:. If graphs G ad H are isomorphic, the there is a permutatio matrix P such that P A(G) P T = A(H) ad hece the matrices A(G) ad A(H) are similar. 2. There are oisomorphic graphs with the same spectrum. See Figure. Figure : A example for two graphs which are ot isomorphic but have the same spectrum. Their commo graph spectrum is 2, 0, 0, 0, 2. 2 Combiatorial Expasio of Graphs For ay d-regular graph G = (V, E), let Γ(v) be the set of eighbors of v, i.e., Γ(v) = {u (u, v) E }. For ay subset S V, let Γ(S) = v S Γ(v) ad Γ (S) = Γ(S) S. Moreoever, for ay set S V we defie S := E(S, S). Defiitio 2.2 (vertex expasio). A graph G with vertices is said to have vertex expasio (K, A) if Γ(S) mi A. S : K If K = /2, the for simplicity we call G a A-expader. Iformally expaders are graphs with the property that every subset (uder some costrait o their size) has may eighbors outside the set. Moreover, we ca use differet ways to study expaders: () Combiatorically, expaders are highly coected graphs, ad to discoect a large part of the graph, oe has to remove may edges; (2) Geometrically, every vertex set has a relatively very large boudary; (3) From the Probabilistic view, expaders are graphs whose behavior is like radom graphs. (4) Algebraically, expaders are the real-symmetric matrix whose first positive eigevalue of the Laplace operator is bouded away from zero. 4
Figure 2: Compariso of the vertex expasio ad the edge expasio of a set of vertices of size 5. Defiitio 2.3 (edge expasio). The edge expasio of a graph G = (V, E) is defied by h(g) := S mi S : V /2. To explai edge expasio, let us see two examples. () If G is ot coected, we choose oe coected compoet as S so that E(S, S) = 0. Therefore h(g) = 0. (2) If G is a complete graph K, the E(S, S) = ( ) ad h(g) = /2. Defiitio 2.4 (expaders). Let d N. A sequece of d-regular graphs {G i } i N of size icreasig with i is a family of expaders if there is a costat ɛ > 0 such that h(g i ) ɛ for all i. Usually, whe speakig of a expader G i, we actually mea a family of graphs {G i } i N, where each graph i {G i } i N is d-regular ad its expasio is lower bouded by ɛ > 0. Observatio 2.5. Ay expader graph is a coected graph. 3 Spectral Expasio vs. Combiatorial Expasio The ext result shows that small spectral expasio implies large vertex expasio. Theorem 2.6 (spectral expasio vertex ( expasio). ) If G has spectral expasio λ, the for all 0 < α <, G has vertex expasio α,. ( α)λ 2 +α Before showig the proof, we itroduce some otatios at first. For ay probability distributio π, the support of π is defied by support(π) = {x : π x > 0}. Defiitio 2.7. Give a probability distributio π, the collisio probability of π is defied to be the probability that two idepedet samples from π are equal, i.e. CP(π) = x π2 x. Lemma 2.8. Let u = (/,..., /) be the uiform distributio. The for every probability distributio π [0, ], we have. CP(π) = π 2 = π u 2 + /. 2. CP(π) / support(π). Proof. () We write π as π = u + (π u) where u (π u). By Pythagorea theorem CP(π) = π 2 = π u 2 + u 2 = π u 2 + /. (2) By Cauchy-Schwarz iequality, we get = x support(π) π x 2 5 support(π) x π 2 x
ad hece CP(π) = x π 2 x support(π). Let us tur to the proof of Theorem 2.6. Proof. Let α. Choose a probability distributio π that is uiform o S ad 0 o the S, i.e. ( ) π =,,...,, 0,..., 0. Note that M is a real symmetric matrix, the M has orthoormal eigevectors v,..., v, the we ca decompose π as i= π i where π i is a costat multiplicity of v i. The CP(π) = / ad by Lemma 2.8 (2), CP(Mπ) support(mπ) = Γ(S). O the other had, by item () of Lemma 2.8 we have CP(Mπ) = Mπ u 2 = Mu + Mπ 2 + + Mπ u 2 = λ 2 π 2 + + λ π 2 ( λ 2 π u 2 = λ 2 CP(π) ) ( = λ 2 ). Hece ad Γ(S) Γ(S) CP(Mπ) ( λ2 ), ( ) λ 2 + λ 2 + ( λ 2 )α = = ( λ 2 α + ( α)λ 2. ) + = λ 2 + ( λ 2 ) / Theorem 2.9 (vertex expasio spectral expasio). Let G be a d-regular graph. For every δ > 0 ad d > 0, there exists γ > 0 such that if G is a d-regular ( + δ)-expader accordig to Defiitio 2.2, the it G has spectral expasio ( γ). Specifically, we ca take γ = Ω(δ 2 /d). Whe talkig about expaders, we ofte mea a family of d-regular graphs satisfyig oe of the followig two equivalet properties: Every graph i the family has spectral expasio λ. Every graph i the family is a ( + δ)-expader for some costat δ. 6
Refereces [Alo86] N. Alo. Eigevalues ad expaders. Combiatorica, 6(2):83 96, 986. [Fri9] Joel Friedma. O the secod eigevalue ad radom walks i radom d-regular graphs. Combiatorica, :33 362, 99. [LPS88] A. Lubotzky, R. Phillips, ad P. Sarak. Ramauja graphs. Combiatorica, 8:26 277, 988. 7