Burning numbers via eigenpolytopes -- Hamming graphs, Johnson graphs, and halved cubes

TL;DR

This paper establishes bounds on the burning number for Hamming, Johnson, and halved cube graphs using eigenpolytope properties and a dynamic search algorithm, extending previous work on hypercube graphs.

Contribution

It introduces new bounds for the burning number of specific graph classes using eigenpolytope analysis and a dynamic search method, generalizing prior hypercube results.

Findings

Lower bounds based on eigenpolytope 1-skeletons

A dynamic search algorithm for unburned vertices

Extension of hypercube burning number results

Abstract

We give lower and upper bounds on the burning number of Hamming graphs, Johnson graphs, and halved cube graphs. For the lower bounds, we use the fact that $1$-skeletons of the eigenpolytopes of these graphs are isomorphic to the original graphs. Then, we present a dynamic search algorithm performed on the eigenpolytope to find an unburned vertex. This idea was originally used by Alon (Discrete Appl.\ Math.,\ 1992), who determined the burning number of the hypercube graphs.