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 $G$.
From: William Watkins [view email]
[v1]
Fri, 12 Jun 2026 23:30:39 UTC (19 KB)
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