On the distance spectrum of the Kronecker product of distance regular graphs

TL;DR

This paper investigates the distance spectrum of the Kronecker product of certain distance regular graphs, providing explicit polynomial expressions and identifying new distance integral graphs, thus advancing spectral graph theory.

Contribution

It determines the distance spectrum for the Kronecker product of Johnson and Hamming graphs and introduces new families of distance integral graphs.

Findings

Derived the exact polynomial for the distance matrix of Johnson and Hamming graphs.

Identified new families of distance integral graphs.

Extended understanding of the distance spectrum in graph products.

Abstract

Consider two simple graphs, G1 and G2, with their respective vertex sets V(G1) and V(G2). The Kronecker product forms a new graph with a vertex set V(G1) X V(G2). In this new graph, two vertices, (x, y) and (u, v), are adjacent if and only if xu is an edge in G1 and yv is an edge in G2. While the adjacency spectrum of this product is known, the distance spectrum remains unexplored. This article determines the distance spectrum of the Kronecker product for a few families of distance regular graphs. We find the exact polynomial, which expresses the distance matrix D as a polynomial of the adjacency matrix, for two distance regular graphs, Johnson and Hamming graphs. Additionally, we present families of distance integral graphs, shedding light on a previously posted open problem given by Indulal and Balakrishnan in (AKCE International Journal of Graphs and Combinatorics, 13(3); 230 to 234, 2016).