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

Document Type : Original Manuscript

Authors

Department of Mathematics, Lorestan University, Khorramabad, Iran.

Abstract

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 $n>2k‎, ‎k\geq1$‎. ‎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 for‎
‎determining the distance spectrum of $H(n,1)$ and show how‎
‎a quotient matrix can contain all distance eigenvalues of‎
‎a graph.‎

Keywords

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