When Are Standard Graph Products Isomorphic?

TL;DR

This paper characterizes when graphs formed by standard products are isomorphic and introduces a new family of non-distance-regular graphs with fewer distance eigenvalues, advancing understanding of graph isomorphism and spectral properties.

Contribution

It provides a complete characterization of isomorphic graph products and identifies a novel family of non-distance-regular graphs with unique spectral features.

Findings

Complete characterization of isomorphic standard graph products

Identification of a new family of non-distance-regular graphs

Graphs with fewer than d+1 distance eigenvalues

Abstract

This article investigates the isomorphism problem for graphs derived from the four standard graph products: Cartesian, Kronecker (direct), strong, and lexicographic product. We provide a complete characterization of all simple connected graphs for which their corresponding products are isomorphic. As a by-product, we identify a novel family of non-distance-regular graphs that possess fewer than d+1 distinct distance eigenvalues, where d represents the diameter of the graph. This result offers a new perspective on Problem 4.3 posed in [2], moving beyond the current approaches.