### 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 labeled 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.

Original language | English |
---|---|

Article number | 315004 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 43 |

Issue number | 31 |

DOIs | |

Publication status | Published - 6 Jul 2010 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modelling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)

### Cite this

*Journal of Physics A: Mathematical and Theoretical*,

*43*(31), [315004]. https://doi.org/10.1088/1751-8113/43/31/315004

**Inductive construction of 2-connected graphs for calculating the virial coefficients.** / Androulaki, E.; Lambropoulou, S.; Economou, Ioannis; Przytycki, J. H.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and Theoretical*, vol. 43, no. 31, 315004. https://doi.org/10.1088/1751-8113/43/31/315004

}

TY - JOUR

T1 - Inductive construction of 2-connected graphs for calculating the virial coefficients

AU - Androulaki, E.

AU - Lambropoulou, S.

AU - Economou, Ioannis

AU - Przytycki, J. H.

PY - 2010/7/6

Y1 - 2010/7/6

N2 - 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 labeled 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.

AB - 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 labeled 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.

UR - http://www.scopus.com/inward/record.url?scp=84943645632&partnerID=8YFLogxK

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U2 - 10.1088/1751-8113/43/31/315004

DO - 10.1088/1751-8113/43/31/315004

M3 - Article

AN - SCOPUS:84943645632

VL - 43

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 31

M1 - 315004

ER -