A generalized Alon-Boppana bound and weak Ramanujan graphs

Transcription

1 A generalized Alon-Boppana bond and weak Ramanjan graphs Fan Chng Abstract A basic eigenvale bond de to Alon and Boppana holds only for reglar graphs. In this paper we give a generalized Alon-Boppana bond for eigenvales of graphs that are not reqired to be reglar. We show that a graph G with diameter k and verte set V, the smallest nontrivial eigenvale λ of the normalized Laplacian L satisfies λ σ c ) k for some constant c where σ = 2 v d v dv / v d2 v and d v denotes the degree of the verte v. We consider weak Ramanjan graphs defined as graphs satisfying λ σ. We eamine the verte epansion and edge epansion of weak Ramanjan graphs and then se the epansion properties among other methods to derive the above Alon-Boppana bond. Introdction The well-known Alon-Boppana bond [8] states that for any d-reglar graph with diameter k, the second largest eigenvale ρ of the adjacency matri satisfies ρ 2 d 2 ) 2 k k. ) A natral qestion is to etend Alon-Boppana bonds for graphs that are irreglar. Hoory [6] showed that for an irreglar graph, the second largest eigenvale ρ of the adjacency matri satisfies ρ 2 d c log r r University of California, San Diego. Research spported in part by AFSOR FA )

2 if the average degree of the graph after deleting a ball of radis r is at least d where r, d > 2. For irreglar graphs, it is often advantageos to consider eigenvales of the normalized Laplacian for deriving varios graph properties. For a graph G, the normalized Laplacian L, defined by L = I D /2 AD /2 where D is the diagonal degree matri and A denotes the adjacency matri of G. One of the main tools for dealing with general graphs is the Cheeger ineqality which relates the least nontrivial eigenvale λ to the Cheeger constant h G : 2h G λ h2 G 2 where h G = min S S) /vols) for S ranging over all verte sbsets with volme vols) = S d no more than half of V d and S) denotes the set of edges leaving S. For k-reglar graphs, we have λ = ρ/k where ρ denotes the second largest eigenvale of the adjacency matri. In general, ρ ma v d v λ ρ min v d v which can be sed to derive a version of the Cheeger ineqality involving ρ which is less effective than 2) for irreglar graphs. In this paper, we will show that for a connected graph G with diameter k, λ is pper bonded by 2) λ σ c k ) 3) for a constant c where σ = 2 v d v dv / v d2 v. The above ineqality will be proved in Section 6. The above bond of Alon-Boppana type improves a reslt of Yong [0] who derived a similar eigenvale bond sing a different method. In [0] the notion of r, d, δ)-robst graphs was considered and it was shown that for a r, d, δ)-robst graph, the least nontrivial eigenvale λ satisfies λ 2d d c ). 4) δ r Here r, d, δ)-robstness means for every verte v and the ball B r v) consisting of all vertices with distance at most r, the indced sbgraph on the complement of B r v) has average degree at least d and v B rv) d2 v/ V \B r v) 2

3 δ. We remark that or reslt in 3) does not reqire the condition of robstness. We define weak Ramanjan graphs to be graphs with eigenvale λ satisfying λ σ 2 5) where σ = 2 v d v dv / v d2 v. To prove the Alon-Boppana bond in 3), it sffices to consider only weak Ramanjan graphs. Weak Ramanjan graphs satisfy varios epansion properties. We will describe several verte-epansion and edge-epansion properties involving λ in Section 3, which will be needed later for proving a diameter bond for weak Ramanjan graphs in Section 4. The diameter bond and related properties of weak Ramanjan graphs are sefl in the proof of the Alon-Boppana bond for general graphs. We will also show that the largest eigenvale λ n of the normalized Laplacian satisfies The proof will be given in Section 7. 2 Preliminaries λ n + σ c ). 6) k For a graph G = V, E), we consider the normalized Laplacian L = I D /2 AD /2 where A denotes the adjacency matri and D denotes the diagonal degree matri with Dv, v) = d v, the degree of v. We assme that there is no isolated verte throghot this paper. For a verte v and a positive integer l, let B l v) denote the ball consisting of all vertices within distance l from v. For an edge {, y} E we say is adjacent to y and write y. Let λ 0 λ... λ n denote eigenvales of L, where n denotes the nmber of vertices in G. It can be checked see [2]) that λ > 0 if G is connected. The Alon-Boppana bond obviosly holds if λ = 0. In the remainder of this paper, we assme G is connected. Let ϕ i denote the orthonormal eigenvector associated with eigenvale λ i. In particlar, ϕ 0 = D /2 / volg) where is the all s vector and 3

4 volg) = v V d v. We can then write g, Lg λ = inf g ϕ 0 g, g ) 2 y f) fy) = inf f D z f 2 z)d z = inf Rf) f D where f ranges over all fnctions satisfying f)d = 0 and the sm y ranges over all nordered pairs {, y} where is adjacent to y. Here Rf) denote the Rayleigh qotient of f, which can be written as follows: f Rf) = f 2 where f 2 = f 2 )d and f = ) 2. f) fy) y For eigenfnction ϕ i, the fnction f i = D /2 ϕ i, called the combinatorial eigenfcntion associated with λ i, satisfies λ i f)d = ) f) fv) 7) v for each verte. In particlar, for f satisfying f)d = 0, we have f, Af λ ) f, Df 8) and f, Af ma i 0 λ i) f, Df. 9) 3 Verte and edge epansions For any sbset S of vertices, there are two types of bondaries. The edge bondary of S, denoted by S) consists of all edges with eactly one endpoint in S. The verte bondary of S, denoted by δs) consists of all vertices not in S bt adjacent to vertices in S. Namely, S) = {{, v} E : S and v S} = ES, S) δs) = { S : v S for some verte v} 4

5 In this section, we will eamine verte epansion and edge epansion relying only on λ. These epansion properties will be needed for deriving diameter bonds for weak Ramanjan graphs which will be sed in or proof of the general Alon-Boppana bond later in Section 6. From the definition of the Cheeger constant, for all verte sbsets S, we have S) vols) h G λ 2 Later in the proofs, we will be interested in the case that vols) is small and therefore we will se the following version. Lemma Let S be a sbset of vertices in G. Then S) vols) λ vols) volg) ). Proof: Sppose f is defined by f = S vols) S vol S) where S denotes the characteristic fnction defined by S v) = if v S and 0 otherwise. The Rayleigh qotient Rf) satisfies λ Rf) = S) vols) volg) vol S). For the epansion of the verte bondary, the Tanner bond [9] for reglar graphs can be generalized as follows. Lemma 2 Let λ = min i 0 λ i. Then for any verte sbset S in a graph, volδs)) vols) λ 2 λ 2 + vols) vols) 0) The proof of the above ineqality is by sing the following discrepancy ineqality as seen in [2]). 5

6 Lemma 3 In a graph G, for two sbset X and Y of vertices, the nmber ex, Y ) = EX, Y ) of edges between X and Y satisfies volx)voly ) ex, Y ) volg) λ volx)voly )volx)voly ) ) volg) where λ = min i 0 λ i. The proof of Lemma 3 follows from 9) and can be fond in [2]. The proof of 2) reslts from ) by setting X = S and Y = S δs). Here we will give a version of the verte-epansion bonds for general graphs which only rely on λ and are independent of other eigenvales. Lemma 4 In a graph G with verte set V and the first nontrivial eigenvale λ, for a sbset S of V with vols δs) ɛvolg) volg)/2, the verte bondary of S satisfies i) volδs)) vols) ii) If /2 λ 2ɛ, then volδs)) vols) Proof: Therefore 2λ λ + 2ɛ 2) λ + 2ɛ) 2. 3) The proof of i) follows from Lemma since volδs)) vols) S δs)) + S) vols) λ ɛ) vols) + volδs) ) + λ ɛ)vols) vols) volδs)) vols) 2λ ɛ) λ ɛ) 2λ λ + 2ɛ To prove ii), we set f = S +γ δs) where γ = λ. Consider g = f c V where c = / f)d volg). By the Cachy-Schwarz ineqality, we have c 2 = volg)) 2 S δs) f)d ) 2 vols δs)) volg)) 2 ɛ volg) f 2 )d f 2 )d. 6

7 Using the ineqality in 8), we have f, Af g, Ag + c 2 volg) γ g, Dg + c 2 volg) = γ f, Df + γ)c 2 volg) γ + ɛ) f, Df = γ + ɛ) vols) + γ 2 volδs)) ). Let es, T ) denote the nmber of ordered pairs, v) where S, v T and {, v} E. Since γ = λ /2, we have Together we have f, Af es, S) + 2γeS, δs)) 2γ)eS, S) + 2γvolS) 2γvolS) volδs)) vols) γ ɛ σ 2 γ + ɛ) γ + 2ɛ) 2 since γ 2ɛ. Recall that weak Ramanjan graphs have eigenvale λ satisfying λ σ 4) where σ = 2 v d v dv / v d2 v. Lemma implies that for S with vols δs)) ɛvolg), volδs)) vols) σ + 2ɛ) 2. For k-reglar Ramanjan graphs with eigenvale λ = 2 k /k, the above ineqality is consistent with the bond volδs)) vols) = δs) S 2 k k + 2ɛ) 2 which is abot k/4 when vols) is small. The factor k/4 in the above ineqality was improved by Kahale [4] to k/2. There are many applications 7

8 see []) that reqire graphs having epansion factor to be ɛ)k. Sch graphs are called lossless epanders. In [], lossless graphs were constrcted eplicitly by sing the zig-zag constrction bt the method for deriving the epansion bonds does not se eigenvales. In this paper, the epansion factor as in Lemma 4 is enogh for or proof later. 4 Weak Ramanjan graphs We recall that a graph is said to be a weak Ramanjan graph as in 4) if where λ σ 2 σ = 2 v d v dv. 5) v d2 v To prove the Alon-Boppana bond, it is enogh to consider only weak Ramanjan graphs. Lemma 5 As defined in 5), σ satisfies 2 d d σ 2 d d where d denotes the average degree in G and d denote the second order degree, i.e., v d = d v and d = v d2 v. n Proof: The proof is mainly by sing the Cachy-Schwarz ineqality. For the pper bond, we note that v σ = 2 d v dv v 2 d2 v d v ) v d2 v v d2 v v = 2 d v ) 2 2 v d2 v v d v vd v ) v d v/ n v d v ) d n 2 d. d 8

9 For the pper bond, we will se the fact that for a, b > and a + b = c, a a + b c b c 2 and therefore d v dv v v v d d v v. n Conseqently, we have σ = 2 v d v dv 2 v d2 v v d v v dv n v d2 v v dv v d v 2 d as desired. We remark that for graphs with average degree at least 20, we have σ < /2 < λ. Theorem Sppose a weak Ramanjan graph G has diameter k. Then for any ɛ > 0, we have 2 log volg) k + ɛ) log σ provided that the volme of G is large, i.e., volg) cσ logσ) /ɛ for some small constant c. Proof: We set t = + ɛ) logvolg)) log σ It sffices to show that for every verte v, the ball B t v) has volme more than volg)/2. Sppose volb t v)) volg)/2. Let s j = volb j)). volg) By part i) of Lemma 4, we have volδb j))) 0.5volB j)) for j t and therefore s j+.5s j. Ths, if j t c logσ ), then s j σ 4 where c is some small constant satisfying c 4log.5).. d 9

10 Now we apply part ii) of Lemma 4 and we have, for j t c logσ ), s j+ s j = volb j+)) volb j )) volδb j))) volb j )) σ + 2s j ) 2 σ + 2σ 4 ). This implies, for l t c logσ ), s l s 0 0<j<l σ + 2s j ) 2 0<j<l Since s 0 /volg) and s l s t /2, we have Hence However, volg) l σ + 2σ 4 ) 2 σ 2l + 2σ 4 ) 2l. σ 2l + 2σ 4 ) 2l. logvolg)) logσ ) + 2σ 4. + ɛ) logvolg)) logσ ) t c logσ ) + logvolg)) logσ ) + 2σ 4 which is a contracdiction for G with volg) large, say, volg) σ 2c log σ /ɛ. Ths we conclde that s t /2 and Theorem is proved. Theorem 2 For a weak Ramanjan graph with diameter k, for any verte v and any l k/4, the ball B l) has volme at most ɛvolg) if k c log ɛ, for some constants c. Proof: We will prove by contradiction. Sppose that for j 0 = k/4, there is a verte with volb v j 0 )) > ɛvolg). Let r denote the largest integer sch that s r = volb r)) volg) > 2. By the assmption, we have r > k/4 and s j0 > ɛ. There are two possibilities: 0

11 Case : r k/2. By part i) of Lemma 4, we have volδb j))) 0.5volB j)) for j k/2 and therefore s j+.5s j. Ths, for j k/2 c log ɛ, we have s j ɛ where c = / log.5. Since k/4 k/2 c log ɛ, we have a contradiction. Case 2: r < k/2. We define s j = volv \ B j)). volg) Ths s j < /2 for all j k/2. We consider two sbcases. Sbcase 2a: Sppose s j ɛ for j k/2. Using Lemma 4, for j where r j k/2, we have s j.5 s j+. Ths, for some j k/2 c log ɛ, we have s j /2 or eqivalently, s j /2. By sing Lemma 4 again, for j j, we have s j+.5s j and therefore for any j j c log ɛ we have s j ɛ. Since j c log ɛ k/2 2c log ɛ k/4, we again have a contradiction to the assmption s j0 ɛ. Sbcase 2b: Sppose s j < ɛ for j k/2 We apply part ii) of Lemma 4 and we have, for j k/2, This implies, for j 2 = k/2, s j2 s k s j s j+ k/2<j k Since s k /volg), we have s j σ + 2ɛ) 2. σ + 2s j ) 2 σ + 2ɛ) k. volg)σ + 2ɛ) k. Since the assmption of this sbcase is s j < ɛ, we have k log n + log ɛ log σ. We now se Lemma 4 and we have, for j = k/2 j r s j volg)σ + 2ɛ) k+2j.

12 Therefore, for some j k/2 log ɛ / log σ, we have s j > /2 which implies r k/2 log ɛ / log σ. Now we se the same argment as in Case ecept shifting r by log ɛ / log σ. For some j r c log ɛ k/2 log ɛ / log σ c log ɛ, we have s j < ɛ. Since log ɛ / log σ +c log ɛ < k/4, this leads to a contradiction and Theorem 2 is proved. 5 Non-backtracking random walks Before we proceed to the proof of the Alon-Boppana bond, we will need some basic facts on non-backtracking random walks. A non-backtracking walk is a seqence of vertices p = v 0, v,..., v t ) for some t sch that v i v i and v i+ v i for i =,..., t 2. The non-backtracking random walk can be described as follows: For i, at the ith step on v i, choose with eqal probability a neighbor of v i where v i, move to and set v i+ =. To simplify notation, we call a nonbacktracking walk an NB-walk. The modified transition probability matri P k, for k = 0,,..., t, is defined by P k, v) if k = 0 P k, v) = wp) if k 6) p P k),v where the weight wp) for an NB-walk p = v 0, v,..., v t ) with t is defined to be wp) = d v0 t i= d v i ) 7) and P,v k) denotes the set of non-backtracking walks from to v. For a walk p = v 0 ) of length 0, we define wp) =. Althogh a non-backtracking random walk is not a Markov chain, it is closely related to an associated Markov chain as we will describe below also see [6]). For each edge {, v} in E, we consider two directed edges, v) and v, ). Let Ê denote the set consisting of all sch directed edges, i.e. Ê = {, v) : {, v} E}. We consider a random walk on Ê with transition 2

13 probability matri P defined as follows: { P, v),, v )) = d v if v = and v 0 otherwise. Let E denote the all s fnction defined on the edge set E as a row vector. From the above definition, we have E P = E. 8) In addition, we define the verte-edge incidence matri B and B for a V and b, c) Ê by { if a = b, Ba, b, c)) = 0 otherwise B b, c), a)) = { if c = a, 0 otherwise. Let V denote all s vector defined on the verte set V. Then V B = E. 9) Althogh P k is not a Markov chain, it is related to the Markov chain determined by P on Ê as follows: Fact : For l. and for the case of l = 0, we have P 0 = I. By combining 9) and 20), we have Fact 2: P l = D BP l B 20) V D P l = E B = V D. 2) Note that V D is jst the degree vector for the graph G. Therefore 2) states that the degree vector is an eigenvector of P l. Using Fact and 2, we have the following: 3

14 Lemma 6 i) For a fied verte and any integer j 0, we have wp) = d 22) d p P j), ii) For a fied verte, we have wp) = I + P P l ) = l + 23) p P j), where denotes the characteristic fnction which assmes vale at and 0 else where. Proof: The proof of 22) and 23) follows from the fact that V D P j ) = V DD BP j B ) = E P j B = E B = V D) and Pj ) = wp) for p P j),. 6 An Alon-Boppana bond for λ Theorem 3 In a graph G = V, E) with diameter k, the first nontrivial eigenvale λ satisfies λ σ c ) k where σ is as defined in 5), provided k c log σ and volg) c σ log σ for some absolte constants c s. Proof: If G is not a weak Ramanjan graph, we have λ σ and we are done. We may assme that G is weak Ramanjan. From the definition of λ, we have λ y f) fy))2 f 2 = Rf) 24) )d where f satisfies f)d = 0. We will constrct an appropriate f satisfying Rf) σ c/k) and therefore serve as an pper bond for λ. We set logvolg)) t = log σ 4

15 and choose ɛ satisfying ɛ σ t cσ k by sing Theorem where σ is as defined in 5). We consider a family of fnctions defined as follows. For a specified verte and an integer l = k/4, we consider a fnction g : V R +, defined by g ) = I + P P ) /2 l )) l = j=0 p P j), ) /2 wp) where P j is as defined in 20) and is treated as a row vector. In other words, g denotes the sqare root of the sm of non-backtracking random walks starting from taking i steps for i ranging from 0 to l. Claim A: d g)d 2 = l j=0 p P l), d wp)d = l + ) d 2 where the weight wp) of a walk p is as defined in 7). Proof of Claim A: From the definition of g and 6), we have 5

16 d g)d 2 = l j=0 = p P l), d wp) d BI + P P l )) d l = D BP i B )d + i= l = BP i B )d + d 2 i= l = i= E P i B )d + d 2 d 2 = l E B )d + d 2 = l + ) d 2. Claim A is proved. Claim B: d g ) g y) ) 2 l + lσ) d 2. y where y denotes the sm ranging over nordered pairs {, y} where is adjacent to y. Proof of Claim B: We will se the following fact for a i, b i > 0. which can be easily checked. ) 2 ai a i b i ) 2 b i 25) i i i 6

17 For a fied verte, we apply Claim B: g ) g y) ) 2 y = y t l r V t l t l t l p P, t) t l wp) p P t),r p =p s P t+),s p P t), p P t), p P t), Using Fact 3, we have d g ) g y) ) 2 t l y d p P,y t) t l wp ) wp) wp )) 2 + wp) wp) d wp) + d 2 d wp) d 2 ) d + p P t), 2 2 p P l), d ) + wp)d ) p P l), )d ) + p P l), wp) d 2 ) d + wp)d ) p P l), wp)d ). d p P l), wp)d ) wp)d ) = l d d 2 d ) + d 2 = l σ) d 2 + = l + lσ ) d 2 This proves Claim B. d 2 Claim C: There is a verte satisfying Rg ) σ ) l + 7

18 Proof of Claim C: Combining Claim A and B, we have d g ) g y) ) 2 y l + lσ ) d 2 l + lσ ) ) d g l + )d 2 = lσ ) d g 2 l + )d 26) Ths we dedce that there is a verte sch that y g ) g y) ) 2 Rg ) = g2 )d lσ l +. 27) We define α v = g v)d d We consider the fnction g defined by = g ) = g ) α Clearly, g satisfies the condition that g )d = 0 Hence, we have λ Rg ) = y g v)d volg) g ) g y) ) 2 g 2 )d y g ) g y) ) 2 = g2 )d αvolg) 2. 28) Note that by the Cachy-Schwarz ineqality, we have ) 2 g )d volb l)) g)d 2. B l) B l) 8

19 and therefore α 2 volb l)) volg) 2 g)d 2. By sbstittion into 28) and sing 35), we have λ Rg ) Rg) volbl)) volg) σ l + σ c ) l + σ l+ volbl)) volg) ) volb l)) + volg) ) 29) 30) 3) The last ineqality follows from Theorem 2 and the choice of ɛ = σ/k. This completes the proof of Theorem 3. 7 A lower bond for λ n If a graph is bipartite, it is known see [2]) that λ i = 2 λ n i for all 0 i n and, in particlar, λ n = 2 λ 0 = 2. If G is not bipartite, it is easy to derive the following lower bond: λ n + /n ) by sing the fact that the trace of L is n. This lower bond is sharp for the complete graph. However if G is not the complete graph, is it possible to derive a better lower bond? The answer is affirmative. Here we give an improved lower bond for λ n. Theorem 4 In a connected graph G = V, E) with diameter k, the largest eigenvale λ n of the normalized Laplacian L of G satisfies λ n + σ c ) 32) k where σ is as defined in 5), provided k c log σ and volg) c σ log σ for some absolte constants c s. Proof: By definition, λ n satisfies λ n y f) fy))2 f 2 = Rf) 33) )d 9

20 for any f : V R. We will constrct an appropriate f sch that Rf) + σ c/γ) by considering the following fnction f : V R +, for a fied verte, defined by { ) t χ Pt ) ) /2 if dist, ) = t l η ) = 0 otherwise where l γ/2. Note that η ) = g ) since we assme that l γ/2. Using the same proof in Claim A, we have Claim A : d η)d 2 = l j=0 p P l), d wp)d = l + ) d 2. Claim B : d η ) η y) ) 2 l + + lσ) d 2. y Proof of Claim B : The proof is qite similar to that of Claim B. For a fied verte, the sm over nordered pair {, y} where y, η ) η y) ) 2 y t l r V t l t l t l p P t),r p =p s P t+),s p P t), p P t), p P t), wp) + wp )) 2 wp) wp) + d wp) + d + 2 d wp) d + 2 ) d 2 p P l), d ) wp)d ) p P l), )d ) p P l), wp)d ) p P l), wp)d ). wp)d ) 20

21 Using Fact 3, we have d η ) η y) ) 2 t l y d p P t), wp) d + 2 ) d d p P l), wp)d ) = l d d + 2 d ) d 2 = l + σ) d 2 = l + lσ ) d 2 d 2 This proves Claim B. Combining Claims A and B, we have d η ) η y) ) 2 y l + lσ ) d 2 l + lσ ) ) d η l + )d 2 = + lσ ) d η 2 l )d 34) Ths we dedce that there is a verte sch that y η ) η y) ) 2 Rη ) = η2 )d + lσ l. 35) We consider the fnction η defined by η ) = η ) α where α v = η v)d d = η v)d volg) 2

22 so that η satisfies the condition that η )d = 0 Hence, we have λ n Rη ) = y η ) η y) ) 2 η 2 )d y η ) η y) ) 2 = η2 )d αvolg) 2 + σ + c l ) volb l)). volg) This completes the proof of Theorem 4. References [] M. Capalbo, O. Reingold, S. Vadhan and A. Wigderson, Randomness condctors and constant-degree lossless epanders, Proceedings of the 34th Annal ACM symposim on Theory of Compting, 2002, [2] F. Chng, Spectral Graph Theory, AMS Pblications, 997, vii+207 pp. [3] J. Friedman, A proof of Alon s second eigenvale conjectre and related problems, Mem. Amer. Math. Soc., ), pp. viii+00 pp. [4] N. Kahale, Eigenvales and epansion of reglar graphs, JACM 42, no ), [5] P. Hall, On representatives of sbsets, J. London Math. Soc. 0 ) 935), [6] S. Hoory, A lower bond on the spectral radis of the niversal cover of a graph, J. Combin. Theory B, ), [7] S. Hoory, N. Linial and A. Wigderson, Epander graphs and their applications, Bll. Amer. Math. Soc., ), [8] A. Nilli, On the second eigenvale of a graph, Discrete Math., 9 99),

23 [9] R. M. Tanner, Eplicit constrction of concentrators from generalized n-gons, SIAM J. Algebraic Discrete Methods, 5 984), [0] S. J. Yong, The weighted spectrm of the niversal cover and an Alon- Boppana reslt for the normalized Laplacian, preprint. 23