Constructing graphs with given spectrum and the spectral radius at most 2

Two graphs are cospectral if their spectra coincide. The set of all graphs that are cospectral to a given graph, including the graph by itself, is the cospectral equivalence class of the graph. We say that a graph is determined by its spectrum, or that it is a DS-graph, if it is a unique graph having that spectrum. Given n reals belonging to the interval [−2,2], we want to find all graphs on n vertices having these reals as the eigenvalues of the adjacency matrix. Such graphs are called Smith graphs. Our search is based on solving a system of linear Diophantine equations. We present several results on spectral characterizations of Smith graphs.