Spectral Properties of the Zeon Combinatorial Laplacian
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866909824691732480 |
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| author | Staples, G. Stacey |
| author_facet | Staples, G. Stacey |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05678 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral Properties of the Zeon Combinatorial Laplacian Staples, G. Stacey Combinatorics Rings and Algebras Spectral Theory 15B33, 15A18, 05C50, 05E15, 81R99 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. |
| title | Spectral Properties of the Zeon Combinatorial Laplacian |
| topic | Combinatorics Rings and Algebras Spectral Theory 15B33, 15A18, 05C50, 05E15, 81R99 |
| url | https://arxiv.org/abs/2502.05678 |