On the number of clusters for planar graphs

Jean-Michel Billiot (1) , Franck Corset (1) , Eric Fontenas (1)

1 LABSAD - Laboratoire de Statistiques et Analyse des Données

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.

Mots clés: Tutte Polynomial; Planar Graphs; Archimedean Graph; Self-Dual Graph; Clusters

Domaines: Mécanique statistique [cond-mat.stat-mech]; Probabilités [math.PR]

Jean-Michel Billiot, Franck Corset, Eric Fontenas. On the number of clusters for planar graphs. 2006. ⟨hal-00078098⟩