ON DETERMINING THE DISTANCE SPECTRUM OF A CLASS OF DISTANCE INTEGRAL GRAPHS

The distance eigenvalues of a connected graph G are the eigenvalues of its distance matrix D(G). A graph is called distance integral if all of its distance eigenvalues are integers. Let n and k be integers with n 2k, k ≥ 1. The bipartite Kneser graph H(n, k) is the graph with the set of all k and n − k subsets of the set [n] = {1, 2, ..., n} as vertices, in which two vertices are adjacent if and only if one of them is a subset of the other. In this paper, we determine the distance spectrum of H(n, 1). Although the obtained result is not new [12], but our proof is new. The main tool that we use in our work is the orbit partition method in algebraic graph theory for finding the eigenvalues of graphs. We introduce a new method for determining the distance spectrum of H(n, 1) and show how a quotient matrix can contain all distance eigenvalues of a graph.