A Generalized Alon-Boppana Bound and Weak Ramanujan Graphs

Fan Chung

doi:10.37236/5933

Abstract

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 $\lambda_1$ of the normalized Laplacian $\mathcal L$ satisfies$$ \lambda_1 \leq 1-\sigma \big(1- \frac c {k} \big)$$ for some constant $c$ where $\sigma = 2\sum_v d_v \sqrt{d_v-1}/\sum_v d_v^2 $ and $d_v$ denotes the degree of the vertex $v$.We consider weak Ramanujan graphs defined as graphs satisfying $ \lambda_1 \geq 1-\sigma$. 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. A corrigendum was added on the 3rd of November 2017.

Highlights

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 √ ρ 2 d−1 1− − . (1)kk the electronic journal of combinatorics 23(3) (2016), #P3.4A natural question is to extend Alon-Boppana bounds for graphs that are irregular
We show that a graph G with diameter k and vertex set V, the smallest nontrivial eigenvalue λ1 of the normalized Laplacian L satisfies c λ1
For the expansion of the vertex boundary, the Tanner bound [9] for regular graphs can be generalized as follows

Summary

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. Hoory [6] showed that for an irregular graph, the second largest eigenvalue ρ of the adjacency matrix satisfies c log r ρ 2 d−1 1−. For k-regular graphs, we have λ1 = 1 − ρ/k where ρ denotes the second largest eigenvalue of the adjacency matrix. To prove the Alon-Boppana bound in (3), it suffices to consider only weak Ramanujan graphs. 4. The diameter bound and related properties of weak Ramanujan graphs are useful in the proof of the Alon-Boppana bound for general graphs.

Preliminaries

Vertex and edge expansions

Weak Ramanujan graphs

Non-backtracking random walks

An Alon-Boppana bound for λ1