Spectral Properties of the Zeon Combinatorial Laplacian: Cycles in Finite Graphs

Abstract

Given a finite simple graph G on m vertices, the zeon combinatorial Laplacian \(\Lambda \) of G is an \(m\times m\) matrix 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.