Graph-induced operators: Hamiltonian cycle enumeration via fermion-zeon convolution

Abstract

Operators are induced on fermion and zeon algebras by the action of adjacency matrices and combinatorial Laplacians on the vector spaces spanned by the graph's vertices. Properties of the algebras automatically give information about the graph's spanning trees and vertex coverings by cycles \& matchings. Combining the properties of operators induced on fermions and zeons gives a fermion-zeon convolution that recovers the number of Hamiltonian cycles in an arbitrary graph. The mathematics underlying the graph-theoretic interpretation of these operators is provided by Kirchhoff's theorem and by the seminal works of Goulden and Jackson and Liu, who established formulas for enumeration of Hamiltonian cycles and paths using determinants and permanents of adjacency matrices.