A unimodular bijection between harmonic vectors of 2-isomorphic graphs

Abstract

Let $G$ and $H$ be connected graphs that are 2-isomorphic. It is known that their Laplacian matrices are congruent by a unimodular matrix $U$. In this paper we show (Thm. \ref{thm:main2}) that $U$ is a bijection between certain spaces of harmonic vectors on the vertices of $G$ and $H$. In particular (Cor. \ref{cor:main1}) if $u$ is a harmonic vector with respect to vertices $c,d$ in $H$ and the 2-isomorphism maps edge $(a,b)$ in $G$ to edge $(c,d)$ in $H$, then $uU$ is a harmonic vector with respect to vertices $a,b$ in $G$.