Relations Between Connected and Self-Avoiding Hikes in Labelled Complete Digraphs

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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