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 its‎‎distance eigenvalues are integers.‎‎Let $n$ and $k$ be integers with $n>2k‎, ‎kgeq1$‎. ‎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 cite{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 for‎‎determining the distance spectrum of $H(n,1)$ and show how‎‎a quotient matrix can contain all distance eigenvalues of‎‎a graph.‎