Spectral Properties of the Zeon Combinatorial Laplacian

Abstract

Given a finite simple graph $G$ on $m$ vertices, the zeon combinatorial Laplacian $\Lambda$ of $G$ is an $m\times m$ graph having entries in the complex zeon algebra $\mathbb{C}\mathfrak{Z}$. It is shown here that if the graph has a unique vertex $v$ of degree $k$, then the Laplacian has a unique zeon eigenvalue $\lambda$ whose scalar part is $k$. Moreover, the canonical expansion of the nilpotent (dual) part of $\lambda$ counts the cycles based at vertex $v$ in $G$. With an appropriate generalization of the zeon combinatorial Laplacian of $G$, all cycles in $G$ are counted by $\Lambda$. Moreover when a generalized zeon combinatorial Laplacian $\Lambda$ can be viewed as a self-adjoint operator on the $\mathbb{C}\mathfrak{Z}$-module of $m$-tuples of zeon elements, it can be interpreted as a quantum random variable whose values reveal the cycle structure of the underlying graph.