An Hopf algebra for counting simple cycles

TL;DR

This paper introduces a Hopf algebra framework for counting simple cycles in directed graphs, providing an exact enumeration formula and a general theorem involving Lie idempotents, advancing combinatorial graph analysis.

Contribution

It constructs an explicit Hopf algebra for counting simple cycles and generalizes the enumeration method using Lie idempotents.

Findings

Derived an exact formula for simple cycle enumeration.

Constructed an explicit Hopf algebra structure.

Established a general theorem involving Lie idempotents.

Abstract

Simple cycles, also known as self-avoiding polygons, are cycles on graphs which are not allowed to visit any vertex more than once. We present an exact formula for enumerating the simple cycles of any length on any directed graph involving a sum over its induced subgraphs. This result stems from an Hopf algebra, which we construct explicitly, and which provides further means of counting simple cycles. Finally, we obtain a more general theorem asserting that any Lie idempotent can be used to enumerate simple cycles.