Enumerative Problems Inspired by Mayer's Theory of Cluster Integrals

Abstract

The basic functional equations for connected and 2-connnected graphs can be traced back to the statistical physicists Mayer and Husimi. They play an essential role in establishing rigorously the virial expansion for imperfect gases. We survey this approach and inspired by these equations, we investigate the problem of enumerating some classes of connected graphs all of whose blocks are contained in a given class B. Included are the species of Husimi graphs (B = "complete graphs"), cacti (B = "unoriented cycles"), and oriented cacti (B = "oriented cycles"). For each of these, we consider the question of their labelled or unlabelled enumeration and of their molecular expansion, according (or not) to their block-size distributions. Résumé Leséquations fonctionnelles de base pour les graphes connexes et 2-connexes remontent aux physiciens Mayer et Husimi. Ces relations sont importantes pouŕ etablir rigoureusement le développement du viriel pour les gaz imparfaits. Nous présentons cette démarche et inspirés par ces relations, nous examinons le problème du dénombrement de quelques classes de graphes connexes dont les blocs sont pris dans une famille B donnée. Cela inclut les graphes de Husimi (B = "graphes complets"), les cactus (B = "polygones"), et les cactus orientés (B = "cycles orientés"). Pour chacune de ces espèces, on s'intéresse au dénombrementétiqueté, ordinaire ou selon la distribution des tailles des blocs, au dénombrement non-étiqueté, et au développement moléculaire.

An Algebraic Representation of Graphs and Applications to Graph Enumeration

“…( ), we obtain Dziobek's recurrence for Cayley's formula [10,24]: in agreement with Husimi's result for this particular case [25] (see also [26]). Also, when the biconnected components are all cycles of length , we recover a particular case of Leroux's result [27]:…”

Об Одном Подходе К Перечислению Помеченных Связных Графов: Обзор Результатов

“…Кактусы служат базовой моделью для ряда задач статистической физики (см. [49]), комбинаторной оптимизации (см. [43]) и теории кодов, исправляющих ошибки (см.…”

“…Перечисление помеченных графов применяется в статистической физике (см. [27][28][29]49]), а также для случайной генерации графов и анализа эффективности алгоритмов (см. [37]), и в теории случайных графов (см.…”

Virial inversion and density functionals

“…Proof. The lemma follows from well-known identities for connected and bi-connected graphs, see for example [Ler04,Far12], we sketch the argument for the reader's convenience. If J ⊂ N is a finite non-empty set, consider the following classes of graphs with vertex set J ∪ {0}:…”