Monotonic normalized heat diffusion for distance-regular graphs with classical parameters of diameter $3$

TL;DR

This paper proves the monotonic normalized heat diffusion property for a specific class of distance-regular graphs with diameter 3, expanding understanding beyond previously known graph classes.

Contribution

It establishes the property for distance-regular graphs with classical parameters of diameter 3, a class not previously confirmed to have this property.

Findings

Proved the property for distance-regular graphs with classical parameters of diameter 3.

Contrasted with known counterexamples and other graph classes where the property holds.

Extended the class of graphs known to satisfy the monotonic normalized heat diffusion property.

Abstract

We prove the monotonic normalized heat diffusion property on distance-regular graphs with classical parameters of diameter $3$. Regev and Shinkar found a Cayley graph for which this property fails. On the other hand, this property has been proved on abelian Cayley graphs, graphs with $3$ distinct eigenvalues and regular bipartite graphs with $4$ distinct eigenvalues by Price, Nica and Kubo-Namba, respectively. A distance regular graph with classical parameters of diameter $3$ has $4$ distinct eigenvalues and is not necessarily bipartite or vertex transitive.