Counting covering cycles

TL;DR

This paper develops a method to count equivalence classes of nonperiodic covering cycles in graphs, relating these counts to determinants, and includes special cases like Euler cycles.

Contribution

It introduces a novel identity linking covering cycle counts to determinants, expanding understanding of cycle enumeration in graphs.

Findings

Derived an explicit formula for counting covering cycles.

Established a determinant-based identity for cycle enumeration.

Connected covering cycle counts to Euler cycles in graphs.

Abstract

We compute the number of equivalence classes of nonperiodic covering cycles of given length in a non oriented connected graph. A covering cycle is a closed path that traverses each edge of the graph at least once. A special case is the number of Euler cycles in the non oriented graph. An identity relating the numbers of covering cycles of any length in a graph to a product of determinants is obtained.