Enumeration of Hamiltonian cycles and paths in a graph

Author:
C. J. Liu

Journal:
Proc. Amer. Math. Soc. **111** (1991), 289-296

MSC:
Primary 05C30; Secondary 05C38, 05C45, 15A15, 15A18

MathSciNet review:
1028288

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Abstract: First, we show that the determinant of a given matrix can be expanded by its principal minors together with a set of arbitrary parameters. The enumeration of Hamiltonian cycles and paths in a graph is then carried out by an algebraic method. Three types of nonalgebraic representation are formulated. The first type is given in terms of the determinant and permanent of a parametrized adjacent matrix. The second type is presented by a determinantal function of multivariables, each variable having domain 0, 1. Formulas of the third type are expressed by spanning trees of subgraphs. When applying the formulas to a complete multipartite graph, one can easily find the results.

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DOI:
https://doi.org/10.1090/S0002-9939-1991-1028288-1

Article copyright:
© Copyright 1991
American Mathematical Society