Cospectral vertices, walk-regular planar graphs and the echolocation problem
Combinatorics
2024-07-22 v2 Spectral Theory
Abstract
We study cospectral vertices on finite graphs in relation to the echolocation problem on Riemannian manifolds. First, We prove a computationally simple criterion to determine whether two vertices are cospectral. Then, we use this criterion in conjunction with a computer search to find minimal examples of various types of graphs on which cospectral but non-similar vertices exist, including minimal walk-regular non-vertex-transitive graphs, which turn out to be non-planar. Moreover, as our main result, we classify all finite 3-connected walk-regular planar graphs, proving that such graphs must be vertex-transitive.
Cite
@article{arxiv.2401.06022,
title = {Cospectral vertices, walk-regular planar graphs and the echolocation problem},
author = {Shi-Lei Kong and Emmett L. Wyman and Yakun Xi},
journal= {arXiv preprint arXiv:2401.06022},
year = {2024}
}
Comments
25 pages, 23 figures. Fixed a gap in the argument, added a toroidal graph example