A spectral characterization of strongly distance-regular graphs with diameter four

TL;DR

This paper provides a spectral characterization of strongly distance-regular graphs with diameter four, revealing that bipartite cases are antipodal, thus advancing understanding of their structural properties.

Contribution

It introduces a spectral criterion for identifying strongly distance-regular graphs with diameter four and shows bipartite cases are antipodal, a novel structural insight.

Findings

Spectral characterization for diameter four strongly distance-regular graphs

Bipartite strongly distance-regular graphs with diameter four are antipodal

Enhanced understanding of the structure of these graphs

Abstract

A graph $G$ with $d+1$ distinct eigenvalues is called strongly distance-regular if $G$ itself is distance-regular, and its distance-$d$ graph $G_d$ is strongly-regular. In this note we provide a spectral characterization of those distance-regular graphs with diameter $d=4$ which are strongly distance-regular. As a byproduct, it is shown that all bipartite strongly distance-regular graphs with such a diameter are antipodal.