On the number of clusters for planar graphs

TL;DR

This paper explores the relationship between the number of clusters per bond in planar graphs and their duals, utilizing the Tutte polynomial to derive factorial moments and applying these findings to various lattice types.

Contribution

It establishes new relations connecting cluster counts in planar graphs and their duals through the Tutte polynomial, incorporating the graphs' coordination numbers.

Findings

Derived factorial moments of cluster counts using Tutte polynomial derivatives.

Presented examples for common planar graphs like square, triangular, and honeycomb lattices.

Discussed implications for various lattice structures such as Archimedean and Laves lattices.

Abstract

The Tutte polynomial is a powerfull analytic tool to study the structure of planar graphs. In this paper, we establish some relations between the number of clusters per bond for planar graph and its dual : these relations bring into play the coordination number of the graphs. The factorial moment measure of the number of clusters per bond are given using the derivative of the Tutte polynomial. Examples are presented for simple planar graph. The cases of square, triangular, honeycomb, Archimedean and Laves lattices are discussed.