Spectral analysis of the affine graph over the finite ring

We apply two methods to the block diagonalization of the adjacency matrix of the Cayley graph defined on the affine group. The affine group will be defined over the finite ring Z/pnZ. The method of irreducible representations will allow us to find nontrivial eigenvalue bounds for two different graphs. One of these bounds will result in a family of Ramanujan graphs. The method of covering graphs will be used to block diagonalize the affine graphs using a Galois group of graph automorphisms. In addition, we will demonstrate the method of covering graphs on a generalized version of the graphs of Lubotzky et al. [A. Lubotzky, R. Phillips, P. Sarnak, Ramanujan graphs, Combinatorica 8 (1988) 261–277].