A generalized Alon-Boppana bound and weak Ramanujan graphs
نویسنده
چکیده
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.
منابع مشابه
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...
متن کاملThe Weighted Spectrum of the Universal Cover and an Alon-boppana Result for the Normalized Laplacian
We provide a lower bound for the spectral radius of the universal cover of irregular graphs in the presence of symmetric edge weights. We use this bound to derive an Alon-Boppana type bound for the second eigenvalue of the normalized Laplacian.
متن کاملThe Measurable Kesten Theorem
We give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d-regular graphs. It follows that the a finite d-regular Ramanujan graph G contains a negligible number of cycles of size less than c log log |G|. We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Through Benjamini-Schramm convergence this leads to the following rigi...
متن کاملGeneralized 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 ...
متن کاملAn 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 ...
متن کامل