On the Expansion of Group-Based Lifts

Transcription

1 On the Expansion of Group-Base Lifts Naman Agarwal 1, Karthekeyan Chanrasekaran 2, Alexanra Kolla 3, an Vivek Maan 4 1 Princeton University, Princeton, Princeton, NJ, USA namana@cs.princeton.eu 2 University of Illinois Urbana-Champaign, Urbana-Champaign, IL, USA karthe@illinois.eu 3 University of Illinois Urbana-Champaign, Urbana-Champaign, IL, USA akolla@illinois.eu 4 University of Illinois Urbana-Champaign, Urbana-Champaign, IL, USA vmaan2@illinois.eu Abstract A k-lift of an n-vertex base graph G is a graph H on n k vertices, where each vertex v of G is replace by k vertices v 1,..., v k an each ege uv in G is replace by a matching representing a bijection π uv so that the eges of H are of the form u i, v πuvi)). Lifts have been investigate as a means to efficiently construct expaners. In this work, we stuy lifts obtaine from groups an group actions. We erive the spectrum of such lifts via the representation theory principles of the unerlying group. Our main results are: 1. A uniform ranom lift by a cyclic group of orer k of any n-vertex -regular base graph G, with the nontrivial eigenvalues of the ajacency matrix of G boune by λ in magnitue, has the new nontrivial eigenvalues boune by λ + O ) in magnitue with probability 1 ke Ωn/2). The probability bouns as well as the epenency on λ are almost optimal. As a special case, we obtain that there is a constant c 1 such that for every k 2 c1n/2, there exists a lift H of every Ramanujan graph by a cyclic group of orer k such that H is almost Ramanujan nontrivial eigenvalues of the ajacency matrix at most O ) in magnitue). We also show how this result leas to a quasi-polynomial time eterministic algorithm to construct almost Ramanujan expaners. 2. There is a constant c 2 such that for every k 2 c2n, there oes not exist an abelian k-lift H of any n-vertex -regular base graph such that H is almost Ramanujan. This can be viewe as an analogue of the well-known no-expansion result for constant egree abelian Cayley graphs. Suppose k 0 is the orer of the largest abelian group that prouces expaning lifts. Our two results highlight lower an upper bouns on k 0 that are tight upto a factor of 3 in the exponent, thus suggesting a threshol phenomenon ACM Subject Classification G.2.2 Graph Theory Keywors an phrases Expaners, Lifts, Spectral Graph Theory Digital Object Ientifier /LIPIcs.APPROX/RANDOM Introuction Expaner graphs have spawne research in pure an applie mathematics uring the last several years, with applications in multiple fiels incluing complexity theory, robust computer networks, error-correcting coes, e-ranomization, compresse sensing an metric A full version of the paper is available at Naman Agarwal, Karthekeyan Chanrasekaran, Alexanra Kolla, an Vivek Maan; license uner Creative Commons License CC-BY Approximation, Ranomization, an Combinatorial Optimization. Algorithms an Techniques APPROX/RAN- DOM 2017). Eitors: Klaus Jansen, José D. P. Rolim, Davi Williamson, an Santosh S. Vempala; Article No. 24; pp. 24:1 24:13 Leibniz International Proceeings in Informatics Schloss Dagstuhl Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany

2 24:2 On the Expansion of Group-Base Lifts embeings [28, 16]. Informally, an expaner is a graph in which every small subset of vertices has a relatively large ege bounary. Most applications are concerne with -regular graphs. The largest eigenvalue of the ajacency matrix of -regular graphs is an is known as a trivial eigenvalue. In case of bipartite -regular graphs, the largest an smallest eigenvalues of their ajacency matrix are an an these are referre to as trivial eigenvalues. The expansion of -regular graphs is etermine by the ifference between an the largest in magnitue) non-trivial eigenvalue of the ajacency matrix, enote λ. Roughly, the smaller λ is, the better the graph expansion. The Alon-Boppana boun [25]) states that λ 2 1 o1) for non-bipartite graphs. Thus, graphs with λ 2 1 are optimal expaners an are calle Ramanujan. A simple probabilistic argument shows the existence of infinite families of expaner graphs [26]. However, constructing such infinite families explicitly has proven to be a challenging an important task. It is easy to construct Ramanujan graphs with a small number of vertices: -regular complete graphs an complete bipartite graphs are Ramanujan. The challenge is to construct an infinite family of -regular graphs that are all Ramanujan, which was first achieve by Lubotzky, Phillips an Sarnak [19] an Margulis [23]. They built Ramanujan graphs from Cayley graphs. All of their graphs are regular, have egrees p + 1 where p is a prime, an their proofs rely on eep number theoretic facts. In two breakthrough papers, Marcus, Spielman, an Srivastava showe the existence of bipartite Ramanujan graphs of all egrees [21, 22]. However they o not provie an efficient algorithm to construct those graphs. Cohen [7] aapte the techniques of [22]to esign an efficient algorithm to construct Ramanujan multi-graphs. A striking result of Frieman [10] an a slightly weaker but more general result of Puer [27], shows that almost every -regular graph on n vertices is very close to being Ramanujan, i.e., for every ɛ > 0, asymptotically almost surely, λ < 2 1 +ɛ. It is still unknown whether the event that a ranom -regular graph is exactly Ramanujan happens with constant probability. Despite a large boy of work on the topic, all attempts to efficiently construct large Ramanujan expaner simple) graphs of any given egree have faile, an exhibiting such a construction remains an intriguing open problem. A combinatorial approach to constructing expaners, initiate by Frieman [9], is to obtain new larger) Ramanujan graphs from smaller ones. In this approach, we start with a base graph which is lifte to obtain a larger graph. Concretely, a k-lift of an n-vertex base-graph G is a graph H on k n vertices, where each vertex u of G is replace by k vertices u 1,..., u k an each ege uv in G is replace by a matching between u 1,..., u k an v 1,..., v k. In other wors, for each ege uv of G there is a permutation π uv of k elements so that the corresponing k eges of H are of the form u i v πuvi). The graph H is a uniformly) ranom lift of G if for every ege uv the bijection π uv is chosen uniformly at ranom from the set S k of permutations of k elements. Since we are focusing on Ramanujan graphs, we will restrict our attention to lifts of -regular graphs. It is easy to see that any lift H of a -regular base-graph G is itself -regular an inherits all the eigenvalues of G. We will refer to the inherite eigenvalues as ol eigenvalues an the rest of the eigenvalues as new eigenvalues. In orer to use the lifts approach for constructing expaners, it is necessary that the lift also inherit the expansion properties of the base graph. Naturally, one hopes that a ranom lift of a Ramanujan graph will also be almost) Ramanujan with high probability. Frieman [9] first stuie the eigenvalues of ranom k-lifts of regular graphs an prove that every new eigenvalue of H is O 3/4 ) with high probability. He conjecture a boun of o1), which woul be tight see, e.g. [14]). Linial an Puer [17] improve Frieman s boun to O 2/3 ). Lubetzky, Suakov an Vu [18] showe that the magnitue

3 N. Agarwal, K. Chanrasekaran, A. Kolla, an V. Maan 24:3 of every nontrivial eigenvalue of the lift is Oλ log ), where λ is the largest in magnitue) nontrivial eigenvalue of the base graph, thus improving on the previous results when G is significantly expaning. Aarrio-Berry an Griffiths [1] further improve the bouns above by showing that every new eigenvalue of H is O ), an Puer [27] prove the nearly-optimal boun of All those results hol with probability tening to 1 as k, thus the orer k of the lift in question nees to be large. Nearly no results were known in the regime where k is boune with respect to the number of noes n of the graph. A relativize version of the Alon-Boppana Conjecture regaring lower-bouning the new eigenvalues of lifts was also recently shown in [12] an [4]. Bilu an Linial [3] were the first to stuy k-lifts of graphs with boune k, an suggeste constructing Ramanujan graphs through a sequence of 2-lifts of a base graph: start with a small -regular Ramanujan graph on some finite number of noes e.g. K +1 ). Every 2-lift operation oubles the number of vertices in the graph. If there is a way to preserve expansion after lifting, then repeating this operation will give large goo expaners of the same boune egree. The authors in [3] showe that if the starting graph G is significantly expaning so that λg) = O log ), then there exists a ranom 2-lift of G that has all its new eigenvalues upper-boune in magnitue by O log 3 ). In a recent breakthrough work, Marcus, Spielman an Srivastava [21] showe that for every bipartite -regular graph G, there exists a 2-lift of G, such that the new eigenvalues achieve the Ramanujan boun of 2 1. But their result still oes not provie an efficient algorithm to fin such lifts. 1.1 Our Results In this work, we stuy the lifts approach to efficiently construct almost Ramanujan expaners of all egrees. We erive these lifts from groups. This is a natural generalization of Cayley graphs. Definition 1 Γ-lift). Let Γ be a group of orer k with enoting the group operation. A Γ-lift of an n-vertex base graph G = V, E) is a graph H = V Γ, E ) obtaine as follows: it has k n vertices, where each vertex u of G is replace by k vertices {u} Γ. For each ege uv of G, we choose an element g uv Γ an replace that ege by a perfect matching between {u} Γ an {v} Γ that is given by the eges u i v j for which g uv i = j. We enote the orer k of the group Γ to be the orer of the lift. We refer to Γ-lifts obtaine using Γ = Z/kZ, the aitive group of integers moulo k, as shift k-lifts. Since every cyclic group of orer k is isomorphic to Z/kZ, we have that Γ-lifts are shift k-lifts whenever Γ is a cyclic group of orer k. A tight connection between the spectrum of Γ-lifts an the representation theory of the unerlying group Γ is known [24, 8]. This connection tells us that the lift incurs the eigenvalues of the base graph, while its new eigenvalues are the union of eigenvalues of a collection of matrices arising from the group elements assigne to the eges an the irreucible representations of the group. We note that this connection has also been recently use in [15] in the context of expansion of lifts, aiming to generalize the results in [22]. In this work, we aress the expansion of Γ-lifts obtaine from cyclic groups an abelian groups. In orer to unerstan the expansion properties of lifts, it suffices to focus on the new eigenvalues of the lifte graph by the above-mentione connection. We present a high probability boun on the expansion of ranom shift k-lifts for boune k. Theorem 2. Let G be a -regular n-vertex graph, where 2 n/3 ln n), with largest in magnitue) non-trivial eigenvalue λ, where λ. Let H be a ranom shift k-lift of G APPROX/RANDOM 17

4 24:4 On the Expansion of Group-Base Lifts with λ new being the largest in magnitue) new eigenvalue of H. Then λ new = Oλ) with probability 1 k e Ωn/2). Moreover, if G is moerately expaning such that λ / log, then λ new λ = O ) with probability 1 k e Ωn/2). We say that a graph is almost Ramanujan if all its non-trivial eigenvalues are boune by O ) in magnitue. By the above result, if the base graph G is Ramanujan, then the ranom shift k-lift will be almost Ramanujan with high probability. Remark 1. In contrast to lifts of orer k, where k when n, the epenency of λ new on λ is necessary for the case of boune k. This has previously been observe by the authors in [3] who gave the following example: Let G be a isconnecte graph on n vertices that consists of n/ + 1) copies of K +1, an let H be a ranom 2-lift of G. Then the largest non-trivial eigenvalue of G is λ = an it can be shown that with high probability, λ new = λ =. Therefore, our eigenvalue bouns are nearly tight. Specializing Theorem 2 for the case of 2-lifts gives the following Corollary which improves upon the multiplicative log factor in the eigenvalue boun that is present in the result of Bilu-Linial [3]. Corollary 3. Let G be a -regular n-vertex graph, where 2 n/3 ln n), with largest in magnitue) non-trivial eigenvalue λ, where λ. Let H be a ranom 2-lift of G with λ new being the largest in magnitue) new eigenvalue of H. Then λ new = Oλ) with probability 1 e Ωn/2). Moreover, if G is moerately expaning such that λ / log, then λ new λ = O ) with probability 1 e Ωn/2). Remark 2. The multiplicative log factor in the eigenvalue boun present in the result of Bilu-Linial [3] arises ue to the use of the converse of the Expaner Mixing Lemma along with an epsilon-net style argument in their analysis. The converse of the Expaner Mixing Lemma is provably tight, so straightforwar use of the converse will inee incur the log factor. We are able to improve the eigenvalue boun by performing a fine-graine analysis of the epsilon-net argument, avoiing irect use of the converse. Lifts base on groups immeiately suggest an algorithm towars builing -regular n- vertex Ramanujan expaners. In orer to escribe this algorithm, we first escribe the brute-force algorithm that follows from the existential result of [21]. The approach is to start with the complete bipartite graph K, an lift the graph log 2 n/2) times. At each stage, we o a brute-force search over the space of all possible 2-lifts an pick the best one i.e.,

5 N. Agarwal, K. Chanrasekaran, A. Kolla, an V. Maan 24:5 one with smallest new maximum eigenvalue in magnitue). However, since a graph V, E) has 2 E possible 2-lifts, it follows that the final lift will be chosen from among 2 n/4 possible 2-lifts, which means that the brute force algorithm will run in time exponential in n. Next, suppose that for every k 2, we are guarantee the existence of a group Γ of orer k such that for every base graph there exists a Γ-lift that has all its new eigenvalues at most 2 1 in magnitue. For example, [5] suggests the possibility that for every k an for every base graph, there exists a shift k-lift that has all new eigenvalues with magnitue at most 2 1. Then a brute force algorithm similar to the one above, woul perform only one lift operation of the base graph K, to create a Γ-lift with n = 2k vertices. This algorithm woul only have to choose the best among k 2 possibilities k ifferent choices of group element per ege of the base graph), which is polynomial in n, the size of the constructe graph here we have assume that is a constant). This motivates the following question: what is the largest possible group Γ that might prouce expaning Γ-lifts? Our next result rules out the existence of large abelian groups that might lea to even slightly) expaning lifts. Theorem 4. For every n-vertex -regular graph G, every real-value ɛ 0, 1/e), an every abelian group Γ of size at least n log 1 k = exp ɛ + log n log 1 eɛ ), all Γ-lifts H of G has a new eigenvalue that is at least ɛ in magnitue. In particular, when k = 2 Ωn), there is no Γ-lift H of any n-vertex -regular graph G all of whose eigenvalues are boune by O ) in magnitue whenever Γ is an abelian group of orer k. Theorem 4 shows that we cannot expect to have arbitrarily large abelian groups with expaning lifts as suggeste in [5]. Remark 3. The first an only known efficient construction of Ramanujan expaner simple graphs are Cayley graphs of certain groups [19]. We observe that a Cayley graph for a group Γ with generator set S can be obtaine as a Γ-lift of the bouquet graph a graph that consists of one vertex with multiple self loops) [20]. Our no-expansion result for abelian groups complements the known result on no-expansion of abelian Cayley graphs [13]. Remark 4. Our Theorems 4 an 2 can be viewe as lower an upper bouns on the largest orer k 0 of an abelian group Γ such that for every n-vertex graph, there exists a Γ-lift for which all new eigenvalues are O ). On the one han, Theorem 2 shows that, for k = 2 On/2), most of the shift k-lifts of a Ramanujan graph have their new eigenvalues to be O ). On the other han, Theorem 4 shows that for k = 2 Ωn), there is no shift k-lift that achieves such eigenvalue guarantees. This suggests a threshol behavior for k 0. We observe that Theorem 2 leas to a eterministic quasi-polynomial time algorithm for constructing almost Ramanujan families of graphs. Theorem 5. There exists an algorithm that runs in time 2 O4 log 2 n) to construct a -regular n-vertex graph such that all its non-trivial eigenvalues are O ) in magnitue. Proof. We use Algorithm 1. We note that the choice of r in the first step ensures that r = O 2 log n). By Theorem 2, there exists a lift G of the base graph G such that APPROX/RANDOM 17

6 24:6 On the Expansion of Group-Base Lifts Algorithm 1 Quasi-polynomial time algorithm to construct expaners of arbitrary size n. 1: Pick an r such that r2 cr/2 = n, for a constant c that appears in the eigenvalue boun in Theorem 2. Do an exhaustive search to fin a -regular graph G on r vertices with λ = O ). 2: For k = 2 cr/2, o an exhaustive search to fin a shift k-lift G of the base graph G with minimum new eigenvalue in magnitue). λg) = O ). Thus, the exhaustive search in the secon step gives a graph G whose non-trivial eigenvalues are O ) in magnitue. In orer to boun the running time, we note that the first step can be implemente to run in time 2 Or2) = 2 O4 log 2 n). To boun the running time of the secon step, we observe that for each ege in G, there are k possible choices. Therefore, the size of the search space is at most k r/2 = 2 cr2 /2 = 2 O3 log 2 n) an for each k-lift, it takes polyn) time to compute λg). Thus, the overall running time of the algorithm is 2 O4 log 2 n). Organization. We give some preliminary efinitions, notations, facts an lemmas in Section 2. We prove Theorem 4 in Section 3. We illustrate the techniques behin proving Theorem 2 by presenting an proving a slightly weaker version of Theorem 2 see Theorem 11) in Section 4. For proofs of the concentration inequality Lemma 12) neee for the weaker version an Theorem 2, we refer the reaer to the full version of the paper [2]. 2 Preliminaries In this section, we efine certain notations an present the neee combinatorial inequalities an facts. Notations. Let G := V, E) be a -regular graph with n vertices. If G is -regular bipartite, we will assume that the bipartition of the vertex set is given by {1,..., n/2}, {n/2+1,..., n}). Let A be the ajacency matrix of G. Since A is a real symmetric matrix, its eigenvalues are also real. Let the eigenvalues of A be λ 1 λ 2... λ n. For a -regular graph G, it is well-known that λ 1 =. If G is bipartite, then λ n = an we efine λ G := max{ λ i : i {2, 3,..., n 1}}. If G is non-bipartite, we efine λ G := max{ λ i : i {2, 3,..., n}}. Thus, λ G enotes the largest in magnitue) non-trivial eigenvalue of G. When G is clear from the context, we will rop the subscript an simply write λ. For subsets S, T V, let ES, T ) be the number of eges uv E with u S an v T. We enote the largest eigenvalue of a matrix M by M an the support of a vector x by Sx). We efine log) to be the log function with base 2. We represent e x by expx). Given a vector x whose coorinates are from {0, ±2 1, ±2 2,..., ±2 i,...} we efine the iaic ecomposition of x as the collection of vectors {2 i u i } i Z where each u i is a vector whose j th coorinate is efine as 1 if x j = 2 i, [u i ] j := 1 if x j = 2 i, 0 otherwise. Lemma 6 Discretization Lemma). Let M R n n be a matrix with iagonal entries being 0.

7 N. Agarwal, K. Chanrasekaran, A. Kolla, an V. Maan 24:7 1. For every x R n with x 1/2 there exists y {0, ±2 1, ±2 2,..., ±2 i,...} n such that x T Mx y T My an y 2 4 x 2. Moreover, each coorinate of x between 2 i an 2 i 1) is roune to either 2 i or 2 i 1) an between 2 i an 2 i 1) is roune to either 2 i or 2 i 1) in y. 2. For every x 1, x 2 R n with x 1, x 2 1/2, there exist y 1, y 2 {0, ±2 1,.., ±2 i,..} n such that x T 1 Mx 2 y T 1 My 2, y x 1 2, y x 2 2 an for b {1, 2} each coorinate of x b between 2 i an 2 i 1) is roune to either 2 i or 2 i 1) an between 2 i an 2 i 1) is roune to either 2 i or 2 i 1) in y b. We nee the following theorem showing that expaners have small iameter in orer to show no-expansion of large abelian lifts. Theorem 7 [6]). The iameter of a -regular graph G with n vertices is at most log n log/λ G ). Lifts. We now state the relevant spectral properties of lifts we erive the spectrum of general group-base lifts in the full version [2]). Some initial easy observations can be mae about the structure of any lift: i) the lifte graph is also regular with the same egree as the base graph an ii) the eigenvalues of the ajacency matrix of the base graph are also eigenvalues of A H. Therefore we call the n eigenvalues of the base graph as the ol eigenvalues an the nk 1) other eigenvalues of A H as the new eigenvalues. We will enote by λ new the largest new eigenvalue of H in magnitue, which we also refer to as the first new eigenvalue for simplicity. Definition 8 Signing). Let G = V, E) be a base graph. Let E f enote an arbitrary orientation of the eges of G an E r enote the reverse orientation. Given a group Γ, a set S an an action of Γ on S as in the Definition 1, we efine a signing of G as a function s : E f E r Γ with the property that if su, v) = g then sv, u) = g 1. We observe that there is a bijection between signings an Γ-lifts. For the purposes of proving the results, we only nee the spectrum of shift k-lifts. For a shift k-lift of a graph G = V, E) with ajacency matrix A, which is given by the signing si, j) = g i,j ) i,j) E, efine the following family of Hermitian matrices A s ω) parameterize by ω where ω is a primitive k-th root of unity: { 0 if Aij = 0, an [A s ω)] ij = ω gi,j if A ij = 1. The following lemma regaring the spectrum of shift k-lifts follows from classic results in representation theory. Lemma 9. Let G = V, E) be a graph an H be a shift k-lift of G with the corresponing signing of the eges si, j) = g i,j ) i,j) E, where g i,j C k. Then the set of eigenvalues of H are given by eigenvalues A s ω)). ω: ω is a primitive k-th root of unity The above simplifies significantly for 2-lifts as note in the next corollary. Corollary 10. When k = 2, the set of eigenvalues of a 2-lift H is given by the eigenvalues of A an the eigenvalues of A s, where A s is the signe ajacency matrix corresponing to the signing s, with entries from {0, 1, 1}. APPROX/RANDOM 17

8 24:8 On the Expansion of Group-Base Lifts 3 No-expansion of Abelian Lifts In this section we show that it is impossible to fin even slightly) expaning graphs using lifts in large abelian groups Γ an thus prove Theorem 4. By Theorem 7, we know that if a graph is an expaner, then it has small iameter. We show that if the size of the abelian) group Γ is large, then all Γ-lifts of any base graph have large iameter, an hence they cannot be expaners. We restate Theorem 4 for convenience. Theorem 4. For every n-vertex -regular graph G, every real-value ɛ 0, 1/e), an every abelian group Γ of size at least n log 1 ɛ k = exp + log n ) log 1, eɛ all Γ-lifts H of G has a new eigenvalue that is at least ɛ in magnitue. In particular, when k = 2 Ωn), there is no Γ-lift H of any n-vertex -regular graph G all of whose eigenvalues are boune by O ) in magnitue whenever Γ is an abelian group of orer k. Proof. We prove the contrapositive. Let Γ be an abelian group of orer k an G = V, E) be a base graph on n-vertices that is -regular. Let e 1,..., e n/2 be an arbitrarily chosen orering of the eges E. Let H be a lift graph obtaine using a Γ-lift. Recall that the signing of the eges of the base graph correspon to group elements, which in turn correspon to permutations of k elements. Let these signing of the eges be σ e ) e EG). Let us efine a layer L i of H to be the set of vertices {v i : v V }. We note that H has k layers. Let us fix an arbitrary vertex v in G. Let enote the iameter of H. Then, for every j {2,..., k} there exists a path of length at most in H from v 1 to a vertex in L j. A layer j is reachable within istance in H iff there exists a walk e 1, e 2,..., e t from v of length t in G such that σ et σ et 1... σ e2 σ e1 1) = j. Thus the set of layers reachable within istance in H is containe in the set S := {σ et... σ e1 1) : e 1,..., e t is a walk from v in G of length t }. Since the group Γ is abelian, S {σe a1 1 σe a σ a n/2 e n/2 1) n/2 i=1 a i } =: T. Since H has k layers, the carinality of S is at least k. The number of integral a i s satisfying n/2 i=1 a i is at most ) n/2)+ n/2) 2 n/2). Therefore, k T n 2 + n 2 ) 2 n 2 2e )) n 2 2e) n 2 e. n Since H has nk vertices, using Theorem 7, we have log nk)/ log/λh)). Thus, if λh) ɛ, then log nk)/ log1/ɛ) an consequently, k 2e) n 2 e log nk log 1 ɛ. Rearranging the terms, we obtain that k 2e) 2 n 1 1 log 1 ɛ ) exp log 1 ɛ log n ) 1 1 log 1 ɛ n log 1 ) exp 4 Expansion of Ranom 2-lifts: Overview ɛ + log n log 1 eɛ ). In this section, we illustrate the main techniques involve in proving Theorem 2 by stating an proving a slightly weaker version, namely Theorem 11. It focuses only on 2-lifts akin

9 N. Agarwal, K. Chanrasekaran, A. Kolla, an V. Maan 24:9 to Corollary 3 an is weaker in comparison to the eigenvalue boun in Corollary 3 by a multiplicative factor of four. The proof of this weaker result captures the main ieas involve in the proof of Theorem 2. Theorem 11. Let G be a -regular n-vertex graph, where 2 n/3 ln n), with largest in magnitue) non-trivial eigenvalue λ, where λ. Let H be a ranom 2-lift of G with λ new being the largest in magnitue) new eigenvalue of H. Then, ) λ new 4λ max λ log, with probability at least 1 e n/2. In orer to prove this theorem, we use the concentration inequality in Lemma 12 recall that for a vector x, its support is enote by Sx)). Lemma 12. Let G be a -regular n-vertex graph, where 2 n/3 ln n), with largest in magnitue) non-trivial eigenvalue λ, where λ. Let H be a ranom 2-lift of G with corresponing signe ajacency matrix A s. The following statements hol with probability at least 1 e n/2 : 1. For all u 1,..., u r {0, ±1} n, an v 1,..., v l {0, ±1} n satisfying I) Su i ) Su j ) = for every i, j [r] an Sv i ) Sv j ) = for every i, j [l], an II) Either Su i ) > n/ 2 for every i [r] with non-zero u i, or Sv i ) > n/ 2 for every i [l] with non-zero v i, we have 2 i u T i )A s 2 j v j ) i j 377 max λ log, r ) Su i ) 2 2i + i=1 ) l λ Sv j ) 2 2j. 3. For all u 1,..., u r {0, ±1} n, an v 1,..., v l {0, ±1} n satisfying I), II) an III) Su i ) > Sv j ) for every i [r], j [l] with non-zero u i, we have 2 i u T i )A s 2 j v j ) i j ) 31 max λ log, r Su i ) 2 2i + i=1 j=1 l Sv j ) 2 2i. We show the concentration inequality in Lemma 12 from Hoeffing s inequality by taking a suitable union boun see the full version of the work [2] for a complete proof). We will now prove Theorem 11 using the lemma above. Our proof strategy resembles the proof strategy in [11]. Proof of Theorem 11. Let s enote the signing corresponing to H an A s enote the signe ajacency matrix. By Corollary 10, the largest in magnitue) new eigenvalue of the lift is λ new = max x R n x T A s x /x T x. To prove an upper boun on λ new, we will boun x T A s x /x T x for all x with high probability. In particular, assuming that the events given by Lemma 12 hol, we will show that x T A s x 4 λ ) x 2. j=1 APPROX/RANDOM 17

10 24:10 On the Expansion of Group-Base Lifts By re-scaling we may assume that the maximum entry of x is less than 1/2 in absolute value. By Lemma 6, there exists a vector y {0, ±2 1, ±2 2,..., ±2 i,...} n such that x T A s x y T A s y an y 2 4 x 2. We will prove a boun on y T A s y for every y {0, ±2 1, ±2 2,..., ±2 i,...} n, which in turn will imply the esire boun on x T A s x. Let us consier the iaic ecomposition of y = i=1 2 i u i obtaine as follows: a coorinate of u i is 1 if the corresponing coorinate of y is 2 i, it is 1 if the corresponing coorinate of y is 2 i, an is zero otherwise. We note that Su i ) Su j ) = for every pair i, j N. Next, we partition the set of vectors u i s base on their support sizes. Let M := {i N : Su i ) n/ 2 } an L := {i N : Su i ) > n/ 2 } we abbreviate M an L for mini an large supports respectively). Corresponingly, efine y M := i M 2 i u i an y L = i L 2 i u i. We note that y = y M + y L, y 2 = y M 2 + y L 2 = i N Su i) 2 2i, an y T A s y y T M A s y M + 2 y T M A s y L + y T LA s y L. We next boun each term in the following three claims. Claim 13. y T M A s y M λ + 8 ) y M 2. Proof. Let y M be a vector obtaine from y M by taking the absolute values of each entry. Then y M 2 = y M 2 an ym T A sy M y M T Ay M. Let J = vvt an J = v v T where v is all ones vector an v is efine as follows: v i = 1 for 1 i n/2 an v i = 1 for n/2 + 1 i n. For non-bipartite graph G, we have y M T Ay M = y M A T n ) ) ) J y M + y M T n J y M λ y M 2 + y M T n J y M. Above, we have use the fact that A nj has the same set of eigenvalues as A except for one the eigenvalue for the matrix A is translate to zero for the matrix A nj. Similarly, for bipartite graphs, we have y M T Ay M = y M A T n J + n ) ) ) J y M + y M T n J y M y M T n J y M ) ) λ y M 2 + y M T n J y M y M T n J y M. Above, we have use the fact that A n J + n J has the same set of eigenvalues as A except for two the largest in magnitue) two eigenvalues for the matrix A are translate to zero for the matrix A n J + n J. It remains to boun y M T n J) y M an y T M n J ) y M. Consier the iaic ecomposition of y M = i M 2 i u i, where the coorinates of u i are the absolute values of the coorinates of u i. ) ) y T M n J y M, y M T n J y M 2 n 2 i Su i ) 2 j Su j ) i M j M:j i i Su i ) 2 i j i M 4 y M 2. j M:j i The secon inequality follows by noting that Su j ) n/ 2 j M

11 N. Agarwal, K. Chanrasekaran, A. Kolla, an V. Maan 24:11 Claim 14. y T LA s y L 2λ ) ) 1012 ) max λ log, y L 2. Proof. By triangle inequality, yla T s y L = 2 i u T i )A s 2 j u j ) i,j L 2 i u i )A s 2 j u j ) + i,j L:i j i,j L:i>j 2 i u i )A s 2 j u j ). We boun each term using the first part of Lemma 12. We now clarify our choice of parameters to apply Lemma 12. For both terms, our choice is r max{i L}, l = r, u i u i if i L an u i 0 if i L, v i = u i for every i [r], where 0 is the all-zeroes vector. We note that the conitions I) an II) of Lemma 12 are satisfie by this choice since every pair Su i ), Su j ) is mutually isjoint an Su i ) > n/ 2 for all i L. Consequently, ) ) λ yla T s y L 754 max λ log, Su i ) 2 2i Su j ) 2 2j i L j L 2λ ) ) ) max λ log, y L 2. Claim 15. y T M A s y L 408 max ) λ λ log, y M ) ) 1012 ) max λ log, y L 2. Proof. By triangle inequality, ym T A s y L = 2 i u T i )A s 2 j u j ) i M,j L 2 i u i )A s 2 j u j ) + i M,j L:i j i M,j L:i>j 2 i u i )A s 2 j u j ). We boun the first an secon terms by the first an secon parts of Lemma 12 respectively. Let 0 be the all-zeroes vector. We now clarify our choice of parameters to apply Lemma 12. For the first term, our choice is r max{i M}, l max{i L}, u i u i if i M an u i 0 if i M, an v i u i if i L an v i 0 if i L. For the secon term, our choice is r max{i L}, l max{i M}, u i u i if i L an u i 0 if i L, an v i u i if i M an v i 0 if i M. The conitions I), II) an III) of Lemma 12 are satisfie for the respective choices since every pair Su i ), Su j ) is mutually isjoint, Su i ) > n/ 2 for all i L an Su i ) > n/ 2 Su j ) for every i L, j M. Consequently, ) ) λ ym T A s y L 377 max λ log, Su i ) 2 2i Su j ) 2 2j i M ) +31 max λ log, Su j ) 2 2j + Su j ) 2 2j j L j M ) ) λ log, y L 2. ) λ 408 max λ log, y M ) max j L APPROX/RANDOM 17

12 24:12 On the Expansion of Group-Base Lifts From the above three claims, we have )) y T A s y λ max λ log, y M 2 + 4λ ) ) 1012 max λ log, y L 2 )) λ max λ log, y 2. Therefore, we have )) x T A s x y T A s y λ max λ log, y 2 )) 4 λ max λ log, x 2. We note that in the above proof, the multiplicative factor of 4 is a by-prouct of the iscretization of x. This can be avoie if we o not iscretize x straightaway, but instea push the iscretization a little eeper into the proof. Inee, we can see that the proof of Claim 13 where we boun y T M A /n)j)y M by λ y M 2 oes not require y M to be a iscretize vector. This is how we are able to prevent the multiplicative factor loss to obtain Theorem 2. References 1 L. Aario-Berry an S. Griffiths. The spectrum of ranom lifts. Preprint arxiv: , N. Agarwal, K. Chanrasekaran, A. Kolla, an V. Maan. On the expansion of group base lifts. Preprint arxiv: , URL: 3 Y. Bilu an N. Linial. Lifts, iscrepancy an nearly optimal spectral gap. Combinatorica, 265): , C. Borenave. A new proof of frieman s secon eigenvalue theorem an its extension to ranom lifts. Preprint arxiv: , K. Chanrasekaran an A. Velingker. Shift lifts preserving ramanujan property. Linear Algebra an its Applications, 529: , F. Chung. Diameters an eigenvalues. Journal of the American Mathematical Society, 22): , M. Cohen. Ramanujan graphs in polynomial time. In 2016 IEEE 57th Annual Symposium on Founations of Computer Science FOCS), pages , R. Feng, J. Kwak, an J. Lee. Characteristic polynomials of graph coverings. Bull. Austal. Math. Soc., 69: , J. Frieman. Relative expaners or weakly relatively ramanujan graphs. Duke Math. J, 118:2003, J. Frieman. A proof of alon s secon eiganvalue conjecture an relate problems. Mem. Amer. Math,Soc, ), J. Frieman, J. Kahn, an E. Szemeréi. On the secon eigenvalue of ranom regular graphs. In Proceeings of the Twenty-first Annual ACM Symposium on Theory of Computing, STOC 89, pages , J. Frieman an D.-E. Kohler. The Relativize Secon Eigenvalue Conjecture of Alon. Preprint arxiv: , J. Frieman, R. Murty, an J. Tillich. Spectral estimates for abelian cayley graphs. J. Comb. Theory Ser. B, 961): , Y. Greenberg. On the spectrum of graphs an their universal coverings. Ph.D Thesis, 1995.

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