{
  "id": "2502.05678",
  "version": "v1",
  "published": "2025-02-08T19:59:59.000Z",
  "updated": "2025-02-08T19:59:59.000Z",
  "title": "Spectral Properties of the Zeon Combinatorial Laplacian",
  "authors": [
    "G. Stacey Staples"
  ],
  "categories": [
    "math.CO",
    "math.RA",
    "math.SP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-08T19:59:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B33",
      "15A18",
      "05C50",
      "05E15",
      "81R99"
    ],
    "keywords": [
      "spectral properties",
      "complex zeon algebra",
      "unique zeon eigenvalue",
      "generalized zeon combinatorial laplacian",
      "finite simple graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}