Abstract
In this paper, inverses of matrices over zeon algebras are discussed and methods of computation are presented. As motivation the zeon combinatorial Laplacian of a simple finite graph is defined, and its inverse is shown to enumerate paths and cycles in its associated graph.
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Notes
The n-particle zeon algebra has been denoted by \({\mathcal {C}\ell _n}^\mathrm{nil}\) in previous works, but that notation is cumbersome for this paper.
In particular, \(\kappa \) is the least positive integer such that \((\mathfrak {D}u)^\kappa =0\).
A forest is an unconnected graph whose connected components are trees.
When \(v_i\) and \(v_j\) have no common neighbors, \(r_{ij}=0\).
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This article is part of the Topical Collection on Proceedings ICCA 12, Hefei, 2020, edited by Guangbin Ren, Uwe Kähler, Rafał Abłamowicz, Fabrizio Colombo, Pierre Dechant, Jacques Helmstetter, G. Stacey Staples, Wei Wang.
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Staples, G.S. Zeon Matrix Inverses and the Zeon Combinatorial Laplacian. Adv. Appl. Clifford Algebras 31, 40 (2021). https://doi.org/10.1007/s00006-021-01152-5
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DOI: https://doi.org/10.1007/s00006-021-01152-5