Block Diagonalization Method for the Covering Graph

Abstract

We present a block diagonalization method for the adjacency matrices of two types of covering graphs. A graph \(Y\) is a covering graph of a base graph \(X\) if there exists an onto graph map \(\pi: Y \to X\) such that for each \(x \in X\) and for each \(y \in \{y \mid \pi(y) = x\}\), the collection of vertices adjacent to \(y\) maps onto the collection of vertices adjacent to \(x \in X\). The block diagonalization method requires the irreducible representations of the Galois group of \(Y\) over \(X\). The first type of covering graph is the Cayley graph over the finite ring \(\mathbb{Z}/p^n\mathbb{Z}\). The second type of covering graph resembles large lattices with vertices \(\mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/n\mathbb{Z}\) for large \(n\). For one lattice, the block diagonalization method allows us to obtain explicit formulas for the eigenvalues of its adjacency matrix. We use these formulas to analyze the distribution of its eigenvalues. For another lattice, the block diagonalization method allows us to find non-trivial bounds on its eigenvalues.