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 current best bounds for codes and apply only to linear codes. However, it is of interest to note that (1) one very simple Alon--Boppana argument yields nontrivial code bound, and (2) our Alon--Boppana argument that equals a current best bound for codes has some hope of improvement. We also improve the bound in sharpest known Alon--Boppana theorem (i.e., when $G$ is a regular tree).