Spectral Properties of the Zeon Combinatorial Laplacian

TL;DR

This paper explores the spectral properties of the zeon combinatorial Laplacian of graphs, revealing how eigenvalues encode cycle structures and can be interpreted as quantum random variables.

Contribution

It introduces a generalized zeon Laplacian that counts all cycles and links spectral properties to quantum interpretations of graph structure.

Findings

Unique eigenvalue with scalar part equal to vertex degree

Nilpotent part counts cycles at a vertex

Laplacian as a quantum random variable

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.