Isospectral points and edges in graph theory

The relationship of isospectral points to symmetrically equivalent points in a graph is described. Many isospectral points are related to symmetrically equivalent verticesvia an equivalence-preserving perturbation. A graph having isospectraledges is examined for clues to assist in finding other such graphs. Two families are found in this manner. Application of equivalence-preserving perturbations to edges that are initially symmetrically equivalent leads to an unlimited number of families of graphs, many with more than one pair of isospectral edges.