On the line graph of a symmetric balanced incomplete block design

1. Introduction. We shall study the relations between an infinite family of finite graphs and the eigenvalues of the corresponding adjacency matrices. All graphs we consider are undirected, finite, with at most one edge joining a pair of vertices, and with no edge joining a vertex to itself. Also, they are all connected and regular (every vertex has the same valence). If G is a graph, its adjacency matrix A = A (G) is given by 1 if i and j are adjacent vertices, ai O otherwise. The line graph L(G) (also called the interchange graph, and the adjoint graph) of a graph G is the graph whose vertices are the edges of G. With two vertices of L(G) adjacent if and only if the corresponding edges of G are adjacent. There have been several investigations in recent years of the extent to which a regular connected graph is characterized by the eigenvalues of its