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Hamiltonian Cycle Enumeration via Fermion-Zeon Convolution

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Abstract

Beginning with a simple graph having finite vertex set V, operators are induced on fermion and zeon algebras by the action of the graph’s adjacency matrix and combinatorial Laplacian on the vector space spanned by the graph’s vertices. When the graph is simple (undirected with no loops or multiple edges), the matrices are symmetric and the induced operators are self-adjoint. The goal of the current paper is to recover a number of known graph-theoretic results from quantum observables constructed as linear operators on fermion and zeon Fock spaces. By considering an “indeterminate” fermion/zeon Fock space, a fermion-zeon convolution operator is defined whose trace recovers the number of Hamiltonian cycles in the graph. This convolution operator is a quantum observable whose expectation reveals the number of Hamiltonian cycles in the graph.

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Notes

  1. Because the zeon algebra is a subalgebra of a Clifford algebra of appropriate signature, it is often denoted by \({\mathcal {C}\ell _{n}}^{\text {nil}}\).

  2. When the edges are ordered pairs, the graph is said to be a directed graph, or digraph.

  3. Let v 0 = (1,…, 1) and observe that L(v 0) = 0.

  4. The normalized trace of \({X}\in \mathcal {L}(V)\) is defined by \( \text {Tr}({X}):=\displaystyle \frac {1}{\dim (V)}\sum \limits _{i=1}^{\dim (V)} \langle \mathbf {v}_{i}\vert {X}\vert \mathbf {v}_{i}\rangle \) for any orthonormal basis {v i : 1 ≤ i ≤ n} of V.

  5. Theorem 4 holds also for directed graphs, provided one eliminates the orientation factor 1/2. In directed graphs, the quantum observable interpretation is lost because adjacency matrices of directed graphs are neither self-adjoint nor normal.

  6. \(\mathfrak {A}=AD\), where A is the usual adjacency matrix and D is a diagonal matrix of zeon generators D = diag(ζ 1,…,ζ n ).

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Acknowledgments

The author thanks Philip Feinsilver for comments and discussions.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL, 62026-1653, USA

    G. Stacey Staples

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  1. G. Stacey Staples
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Correspondence to G. Stacey Staples.

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Staples, G.S. Hamiltonian Cycle Enumeration via Fermion-Zeon Convolution. Int J Theor Phys 56, 3923–3934 (2017). https://doi.org/10.1007/s10773-017-3381-z

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