Inductive Construction of 2-Connected Graphs for Calculating the Virial Coefficients

TL;DR

This paper presents a systematic inductive method for constructing all simple 2-connected graphs with n vertices, facilitating the calculation of virial coefficients in the theory of non-ideal gases.

Contribution

It introduces a novel inductive construction technique for 2-connected graphs using edge subdivision and vertex addition, relevant for computing virial coefficients.

Findings

Provides a recursive method for generating all 2-connected graphs

Maintains correspondence between graphs and particle clusters

Enables calculation of virial coefficients from smaller graphs

Abstract

In this paper we give a method for constructing systematically all simple 2-connected graphs with n vertices from the set of simple 2-connected graphs with n-1 vertices, by means of two operations: subdivision of an edge and addition of a vertex. The motivation of our study comes from the theory of non-ideal gases and, more specifically, from the virial equation of state. It is a known result of Statistical Mechanics that the coefficients in the virial equation of state are sums over labelled 2-connected graphs. These graphs correspond to clusters of particles. Thus, theoretically, the virial coefficients of any order can be calculated by means of 2-connected graphs used in the virial coefficient of the previous order. Our main result gives a method for constructing inductively all simple 2-connected graphs, by induction on the number of vertices. Moreover, the two operations we are using maintain the correspondence between graphs and clusters of particles.