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arXiv · 2606.15018

A unimodular bijection between harmonic vectors of 2-isomorphic graphs

William Watkins
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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 $G$.

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William Watkins. 2026-06-12. A unimodular bijection between harmonic vectors of 2-isomorphic graphs. https://arxiv.org/abs/2606.15018

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