Abstract
A walk in a directed graph is defined as a finite sequence of contiguous edges. Seeing the edges as indeterminates, walks are investigated as monomials and endowed with a partial order that extends to possibly unconnected objects called hikes. Analytical transformations of the weighted adjacency matrix reveal a relation between walks and self-avoiding hikes, giving rise to interesting combinatorial properties such as an expression of the number of ways to travel a walk in function of its self-avoiding divisors.
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Acknowledgments
The authors are grateful to Pierre-Louis Giscard for his explanations on the poset structure of hikes, and to an anonymous referee for helpful comments which helped improve this paper.
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Espinasse, T., Rochet, P. Relations Between Connected and Self-Avoiding Hikes in Labelled Complete Digraphs. Graphs and Combinatorics 32, 1851–1871 (2016). https://doi.org/10.1007/s00373-016-1696-9
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DOI: https://doi.org/10.1007/s00373-016-1696-9