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≥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 \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

References

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Journal of Algebraic Systems
Volume 10, Issue 2
January 2023
Pages 299-308

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