A generalized Alon-Boppana bound and weak Ramanujan graphs

A basic eigenvalue bound due to Alon and Boppana holds only for regular graphs. In this paper we give a generalized Alon-Boppana bound for eigenvalues of graphs that are not required to be regular. We show that a graph G with diameter k and vertex set V , the smallest nontrivial eigenvalue λ1 of the normalized Laplacian L satisfies λ1 6 1− σ ( 1− c k ) for some constant c where σ = 2 ∑ v dv √ dv − 1/ ∑ v d 2 v and dv denotes the degree of the vertex v. We consider weak Ramanujan graphs defined as graphs satisfying λ1 > 1 − σ. We examine the vertex expansion and edge expansion of weak Ramanujan graphs and then use the expansion properties among other methods to derive the above Alon-Boppana bound.


Introduction
The well-known Alon-Boppana bound [8] states that for any d-regular graph with diameter k, the second largest eigenvalue ρ of the adjacency matrix satisfies A natural question is to extend Alon-Boppana bounds for graphs that are irregular. Hoory [6] showed that for an irregular graph, the second largest eigenvalue ρ of the adjacency matrix satisfies if the average degree of the graph after deleting a ball of radius r is at least d where r, d > 2.
For irregular graphs, it is often advantageous to consider eigenvalues of the normalized Laplacian for deriving various graph properties. For a graph G, the normalized Laplacian L, defined by where D is the diagonal degree matrix and A denotes the adjacency matrix of G. One of the main tools for dealing with general graphs is the Cheeger inequality which relates the least nontrivial eigenvalue λ 1 to the Cheeger constant h G : where h G = min S |∂(S)|/vol(S) for S ranging over all vertex subsets with volume vol(S) = u∈S d u no more than half of u∈V d u and ∂(S) denotes the set of edges leaving S. For k-regular graphs, we have λ 1 = 1 − ρ/k where ρ denotes the second largest eigenvalue of the adjacency matrix. In general, which can be used to derive a version of the Cheeger inequality involving ρ which is less effective than (2) for irregular graphs.
In this paper, we will show that for a connected graph G with diameter k, λ 1 is upper bounded by The above inequality will be proved in Section 6.
The above bound of Alon-Boppana type improves a result of Young [10] who derived a similar eigenvalue bound using a different method. In [10] the notion of (r, d, δ)-robust graphs was considered and it was shown that for a (r, d, δ)-robust graph, the least nontrivial eigenvalue λ 1 satisfies Here (r, d, δ)-robustness means for every vertex v and the ball B r (v) consisting of all vertices with distance at most r, the induced subgraph on the complement of B r (v) has average degree at least d and v ∈Br(v) d 2 v /|V \ B r (v)| δ. We remark that our result in (3) does not require the condition of robustness.
We define weak Ramanujan graphs to be graphs with eigenvalue λ 1 satisfying To prove the Alon-Boppana bound in (3), it suffices to consider only weak Ramanujan graphs. Weak Ramanujan graphs satisfy various expansion properties. We will describe several vertex-expansion and edge-expansion properties involving λ 1 in Section 3, which will be needed later for proving a diameter bound for weak Ramanujan graphs in Section 4. The diameter bound and related properties of weak Ramanujan graphs are useful in the proof of the Alon-Boppana bound for general graphs.
We will also show that the largest eigenvalue λ n−1 of the normalized Laplacian satisfies The proof will be given in Section 7.

Preliminaries
For a graph G = (V, E), we consider the normalized Laplacian where A denotes the adjacency matrix and D denotes the diagonal degree matrix with D(v, v) = d v , the degree of v. We assume that there is no isolated vertex throughout this paper. For a vertex 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 {x, y} ∈ E we say x is adjacent to y and write x ∼ y. Let λ 0 λ 1 . . . λ n−1 denote eigenvalues of L, where n denotes the number of vertices in G. It can be checked (see [2]) that λ 1 > 0 if G is connected. The Alon-Boppana bound obviously holds if λ 1 = 0. In the remainder of this paper, we assume G is connected.
Let ϕ i denote the orthonormal eigenvector associated with eigenvalue λ i . In particular, ϕ 0 = D 1/2 1/ vol(G) where 1 is the all 1's vector and vol(G) = v∈V d v . We can then write where f ranges over all functions satisfying u f (u)d u = 0 and the sum x∼y ranges over all unordered pairs {x, y} where x is adjacent to y. Here R(f ) denote the Rayleigh quotient of f , which can be written as follows: For eigenfunction ϕ i , the function f i = D −1/2 ϕ i , called the combinatorial eigenfucntion associated with λ i , satisfies for each vertex u. In particular, for f satisfying u f (u)d u = 0, we have and 3 Vertex and edge expansions In this section, we will examine vertex expansion and edge expansion relying only on λ 1 . These expansion properties will be needed for deriving diameter bounds for weak Ramanujan graphs which will be used in our proof of the general Alon-Boppana bound later in Section 6.
From the definition of the Cheeger constant, for all vertex subsets S, we have Later in the proofs, we will be interested in the case that vol(S) is small and therefore we will use the following version.

Lemma 1. Let S be a subset of vertices in G.
Then Proof. Suppose f is defined by The Rayleigh quotient R(f ) satisfies For the expansion of the vertex boundary, the Tanner bound [9] for regular graphs can be generalized as follows.
The proof of the above inequality is by using the following discrepancy inequality (as seen in [2]).
Lemma 3. In a graph G, for two subset X and Y of vertices, the number e(X, Y ) = |E(X, Y )|of edges between X and Y satisfies The proof of Lemma 3 follows from (9) and can be found in [2]. The proof of (12) results from (11) by setting X = S and Y = S ∪ δ(S).
Here we will give a version of the vertex-expansion bounds for general graphs which only rely on λ 1 and are independent of other eigenvalues.
Lemma 4. In a graph G with vertex set V and the first nontrivial eigenvalue the electronic journal of combinatorics 23 (2016), #P3.4 Proof. The proof of (i) follows from Lemma 1 since vol(δ(S)) vol(S) . By the Cauchy-Schwarz inequality, we have Using the inequality in (8), we have Let e(S, T ) denote the number of ordered pairs (u, v) where u ∈ S, v ∈ T and {u, v} ∈ E.
the electronic journal of combinatorics 23 (2016), #P3.4 Recall that weak Ramanujan graphs have eigenvalue λ 1 satisfying For k-regular Ramanujan graphs with eigenvalue which is about k/4 when vol(S) is small. The factor k/4 in the above inequality was improved by Kahale [4] to k/2. There are many applications (see [1]) that require graphs having expansion factor to be (1 − )k. Such graphs are called lossless expanders. In [1], lossless graphs were constructed explicitly by using the zig-zag construction but the method for deriving the expansion bounds does not use eigenvalues. In this paper, the expansion factor as in Lemma 4 is enough for our proof later.

Weak Ramanujan graphs
We recall that a graph is said to be a weak Ramanujan graph as in (14) if To prove the Alon-Boppana bound, it is enough to consider only weak Ramanujan graphs.
Lemma 5. As defined in (15), σ satisfies whered denotes the average degree in G andd denote the second order degree, i.e., the electronic journal of combinatorics 23 (2016), #P3.4 Proof. The proof is mainly by using the Cauchy-Schwarz inequality. For the upper bound, we note that For the upper bound, we will use the fact that for a, b > 1 and a + b = c, Consequently, we have We remark that for graphs with average degree at least 20, we have σ < 1/2 < λ 1 .
Theorem 6. Suppose a weak Ramanujan graph G has diameter k. Then for any > 0, we have provided that the volume of G is large, i.e., vol(G) cσ log(σ) / for some small constant c.
Proof. We set It suffices to show that for every vertex v, the ball B t (v) has volume more than vol(G)/2. Suppose vol(B t (v)) vol(G)/2. Let the electronic journal of combinatorics 23 (2016), #P3.4 By part (i) of Lemma 4, we have vol(δ(B u (j))) 0.5vol(B u (j)) for j t − 1 and therefore s j+1 1.5s j . Thus, if j t − c 1 log(σ −1 ), then s j σ 4 where c 1 is some small constant satisfying c 1 4(log 1.5) −1 . Now we apply part (ii) of Lemma 4 and we have, for j t − c 1 log(σ −1 ), .
Theorem 7. For a weak Ramanujan graph with diameter k, for any vertex v and any l k/4, the ball B u (l) has volume at most vol(G) if k c log −1 , for c = 1/(log 1.5).
Proof. We will prove by contradiction. Suppose that for j 0 = k/4 , there is a vertex u with vol(B u (j 0 )) > vol(G). Let r denote the least integer such that By the assumption, we have r > k/4 and s j 0 > . There are two possibilities: Case 1: r k/2. By part (i) of Lemma 4, we have vol(δ(B u (j))) 0.5vol(B u (j)) for j k/2 and therefore s j+1 1.5s j . Thus, for j k/2 − c 1 log −1 , we have s j where c 1 = 1/ log 1.5. Since k/4 k/2 − c 1 log −1 , we have a contradiction. Case 2: r < k/2. We defines j+1 = vol(V \ B u (j)) vol(G) .
Subcase 2a: Supposes j for some j k/2. Using Lemma 4, for j where r j k/2, we haves j 1.5s j+1 . Thus, for some j 1 k/2 − c 1 log −1 , we haves j 1/2 or equivalently, s j 1/2. By using Lemma 4 again, for j j 1 , we have s j+1 1.5s j and therefore for any j j 1 − c 1 log −1 we have s j . Since j 1 − c 1 log −1 k/2 − 2c 1 log −1 k/4, we again have a contradiction to the assumption s j 0 .
Subcase 2b: Supposes j < for all j k/2 We apply part (ii) of Lemma 4 and we have, for j k/2, This implies, for j 2 = k/2 , Sinces k 1/vol(G), we haves Since the assumption of this subcase iss j 1 < , we have k log n + log −1 log σ −1 .

Non-backtracking random walks
Before we proceed to the proof of the Alon-Boppana bound, we will need some basic facts on non-backtracking random walks.
A non-backtracking walk is a sequence of vertices p = (v 0 , v 1 , . . . , v t ) for some t such that v i−1 ∼ v i and v i+1 = v i−1 for i = 1, . . . , t − 2. The non-backtracking random walk can be described as follows: For i 1, at the ith step on v i , choose with equal probability a neighbor u of v i where u = v i−1 , move to u and set v i+1 = u. To simplify notation, we call a non-backtracking walk an NB-walk. The modified transition probability matrixP k , for k = 0, 1, . . . , t − 1, is defined bỹ where the weight w(p) for an NB-walk p = (v 0 , v 1 , . . . , v t ) with t 1 is defined to be and P (k) u,v denotes the set of non-backtracking walks from u to v. For a walk p = (v 0 ) of length 0, we define w(p) = 1.
Although 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 {u, v} in E, we consider two directed edges (u, v) and (v, u). LetÊ denote the set consisting of all such directed edges, i.e.Ê = {(u, v) : {u, v} ∈ E}. We consider a random walk onÊ with transition probability matrix P defined as follows: Let 1 E denote the all 1's function defined on the edge set E as a row vector. From the above definition, we have In addition, we define the vertex-edge incidence matrix B and B * for a ∈ V and (b, c) ∈Ê by B (a, (b, c) Let 1 V denote all 1's vector defined on the vertex set V . Then AlthoughP k is not a Markov chain, it is related to the Markov chain determined by P onÊ as follows: and for the case of l = 0, we haveP 0 = I. By combining (19) and (20), we have Fact 2: Note that 1 V D is just the degree vector for the graph G. Therefore (21) states that the degree vector is an eigenvector ofP l . Using Fact 1 and 2, we have the following: Lemma 8.
(i) For a fixed vertex x and any integer j 0, we have (ii) For a fixed vertex u, we have x p∈P Proof. The proof of (22) and (23) follows from the fact that and 1 uPj (x) = w(p) for p ∈ P (j) u,x .
6 An Alon-Boppana bound for λ 1 Theorem 9. In a graph G = (V, E) with diameter k, the first nontrivial eigenvalue λ 1 satisfies Proof. If G is not a weak Ramanujan graph, we have λ 1 1 − σ and we are done. We may assume that G is weak Ramanujan.
From the definition of λ 1 , we have where f satisfies x f (x)d x = 0. We will construct an appropriate f satisfying R(f ) 1 − σ(1 − c/k) and therefore serve as an upper bound for λ 1 .
We consider a family of functions defined as follows. For a specified vertex u and an integer l = k/4 , we consider a function g u : V → R + , defined by whereP j is as defined in (20) and 1 u is treated as a row vector. In other words, g u denotes the square root of the sum of non-backtracking random walks starting from u taking i steps for i ranging from 0 to l.
where the weight w(p) of a walk p is as defined in (17).

Proof of Claim
Claim A is proved.
where x∼y denotes the sum ranging over unordered pairs {x, y} where x is adjacent to y.

Proof of Claim B:
We will use the following fact for which can be easily checked. For a fixed vertex u, we apply Claim B: Combining Claim A and B, we have Thus we deduce that there is a vertex u such that We define We consider the function g u defined by Clearly, g u satisfies the condition that Hence, we have Note that by the Cauchy-Schwarz inequality, we have the electronic journal of combinatorics 23 (2016), #P3.4 and therefore α 2 u vol(B u (l)) vol(G) 2 x g 2 u (x)d x .

A lower bound for λ n−1
If a graph is bipartite, it is known (see [2]) that λ i = 2 − λ n−i−1 for all 0 i n − 1 and, in particular, λ n−1 = 2 − λ 0 = 2. If G is not bipartite, it is easy to derive the following lower bound: λ n−1 1 + 1/(n − 1) by using the fact that the trace of L is n. This lower bound is sharp for the complete graph. However if G is not the complete graph, is it possible to derive a better lower bound? The answer is affirmative. Here we give an improved lower bound for λ n−1 .
Theorem 10. In a connected graph G = (V, E) with diameter k, the largest eigenvalue λ n−1 of the normalized Laplacian L of G satisfies 1/2 and k(1.5) k σ −1 where d v denotes the degree of the vertex v with minimum degree at least 2.
Proof. By definition, λ n−1 satisfies for any f : V → R. Hence, we have This completes the proof of Theorem 10.