A Generalized Alon-Boppana Bound and Weak Ramanujan Graphs

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 √ d...

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 √ d...

Generalized Alon--Boppana Theorems and Error-Correcting Codes

In this paper we describe several theorems that give lower bounds on the second eigenvalue of any quotient of a given size of a fixed graph, G. These theorems generalize Alon-Boppana type theorems, where G is a regular (infinite) tree. When G is a hypercube, our theorems give minimum distance upper bounds on linear binary codes of a given size and information rate. Our bounds at best equal the ...

An Alon-Boppana Type Bound for Weighted Graphs and Lowerbounds for Spectral Sparsification

We prove the following Alon-Boppana type theorem for general (not necessarily regular) weighted graphs: if G is an n-node weighted undirected graph of average combinatorial degree d (that is, G has dn/2 edges) and girth g > 2d + 1, and if λ1 ≤ λ2 ≤ · · ·λn are the eigenvalues of the (non-normalized) Laplacian of G, then λn λ2 ≥ 1 + 4 √ d −O ( 1 d 5 8 ) (The Alon-Boppana theorem implies that if ...

Ramanujan graphs