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

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Kogani Roya; Mirafzal Seyed Morteza

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Journal Paper

Paper Information

Journal: Journal of Algebraic Systems Year:2023 | Volume:10 | Issue:2 Page(s): 299-308

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Title

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

Author(s)

Kogani Roya | Mirafzal Seyed Morteza | Issue Writer Certificate

Keywords

Distance matrix

distance spectrum

orbit partition

bipartite Kneser graph

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.

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