Spectral Properties of the Zeon Combinatorial Laplacian
G. Stacey Staples
TL;DR
The paper develops and analyzes the zeon combinatorial Laplacian, a matrix with entries in the complex zeon algebra, to count graph cycles and extract cycle-based structure. It proves that if a graph has a unique vertex of degree $k$, the zeon Laplacian has a spectrally simple eigenvalue with scalar part $k$, whose dual part encodes cycles based at that vertex, and it provides explicit expressions for the eigenvalue and eigenvector in terms of paths and PWICs. By extending to generalized labeled Laplacians and to symmetric and $q$-weighted variants, the work shows how all cycles can be counted and interpreted via matrix exponentials of the nilpotent adjacency, with a quantum-observable interpretation for the symmetric case. The framework thus links zeon algebra, spectral theory, and cycle enumeration in graphs, offering potential quantum-information applications and avenues for further exploration in time-evolving graphs and colored-graph problems.
Abstract
Given a finite simple graph $G$ on $m$ vertices, the zeon combinatorial Laplacian $Λ$ 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 $λ$ whose scalar part is $k$. Moreover, the canonical expansion of the nilpotent (dual) part of $λ$ 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 $Λ$. Moreover when a generalized zeon combinatorial Laplacian $Λ$ 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.