On the number of circuits in random graphs

TL;DR

This paper uses statistical mechanics methods to estimate the number of long circuits in graphs, providing both an approximate counting algorithm and theoretical insights into random graph ensembles.

Contribution

It introduces a non-rigorous statistical mechanics approach for counting long circuits and offers an approximate counting procedure applicable to various graphs.

Findings

Proposes an approximate counting algorithm for long circuits.

Reproduces known results on the number of circuits in random graphs.

States new conjectures on circuit counts in graph ensembles.

Abstract

We apply in this article (non rigorous) statistical mechanics methods to the problem of counting long circuits in graphs. The outcomes of this approach have two complementary flavours. On the algorithmic side, we propose an approximate counting procedure, valid in principle for a large class of graphs. On a more theoretical side, we study the typical number of long circuits in random graph ensembles, reproducing rigorously known results and stating new conjectures.