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Spectral properties of the exponential distance matrix

### Steve Butler, Elizabeth Cooper, Aaron Li, Kate Lorenzen and Zoë Schopick

Vol. 15 (2022), No. 5, 739–762
##### Abstract

Given a graph $G$, the exponential distance matrix is defined entrywise by letting the $\left(u,v\right)$-entry be ${q}^{\mathrm{dist}\left(u,v\right)}$, where $\mathrm{dist}\left(u,v\right)$ is the distance between the vertices $u$ and $v$ with the convention that if vertices are in different components, then ${q}^{\mathrm{dist}\left(u,v\right)}=0$. We will establish several properties of the characteristic polynomial (spectrum) for this matrix, give some families of graphs which are uniquely determined by their spectrum, and produce cospectral constructions.

##### Keywords
exponential distance matrix, spectral graph theory, Cartesian product, cospectral graphs
Primary: 05C50
##### Milestones
Received: 13 October 2019
Revised: 18 January 2022
Accepted: 31 January 2022
Published: 3 March 2023

Communicated by Kenneth S. Berenhaut
##### Authors
 Steve Butler Department of Mathematics Iowa State University Ames, IA United States Elizabeth Cooper Brooklyn, NY United States Aaron Li University of Minnesota Minneapolis, MN United States Kate Lorenzen Linfield University McMinville, OR United States Zoë Schopick Tucson, AZ United States