Other Formulas for the Ree-Hoover and Mayer Weights of Families of 2-Connected Graphs

We study graph weights which naturally occur in Mayer’s theory and Ree-Hoover’s theory for the virial expansion in the context of an imperfect gas. We pay particular attention to the Mayer weight and Ree-Hoover weight of a 2-connected graph in the case of the hard-core continuum gas in one dimension. These weights are calculated from signed volumes of convex polytopes associated with the graph. In the present paper, we use the method of graph homomorphisms, to develop other explicit formulas of Mayer weights and Ree-Hoover weights for infinite families of 2-connected graphs.


Introduction
Before discussing our subject, we first present some preliminary notions on the theory of graphs drawn from among others [1] [2] [3].

Preliminary Notions on the Theory of Graphs
there is a chain from v to w.
Any graph breaks down uniquely as a disjoint union of connected graphs. Definition 6. On the set V of the vertices of the simple graph ( ) , we define the relation of equivalence: ṽ w ⇔ there is a chain v to w in g. Let 1 2 , , , k V V V  the equivalence classes of ~ and let's say, for 1 i k the subgraph of g generated by i V . These simple graphs i g , that we call the connected components of g, are related (see Figure 1 with connected components are circled). Definition 7. A cutpoint (or articulation point) of a connected graph c is a vertex of c whose removal yields a disconnected graph. Definition 8. A connected graph is called 2-connected if it has no cutpoint (see Figure 2).
In the present paper, we study Graph weights in the context of a non-ideal gas The interest of this sequence in statistical mechanics comes from the fact that the pressure P of the system is given by its exponential generating function as follows (see [6]): where χ denote the characteristic function ( ( ) 1 P χ = , if P is true and 0, otherwise).
The main goal of the present paper is to give new explicit formulas for the Mayer and Ree-Hoover weights of certain infinite families of graphs in the context of the hard core continuum gas, defined by (7), in dimension  [14].

A. Kaouche Journal of Applied Mathematics and Physics
In Section 2, we look at the case of the hard-core continuum gas in one dimension in which the Mayer weight turns out to be a signed volume of a convex polytope ( ) c  naturally associated with the graph c. A decomposition of the polytope ( ) c  into a certain number of simplices is utilised. This method was introduced in [6] and was adapted in [1] [5] to the context of Ree-Hoover weights and is called the method of graph homomorphisms. The explicit computation of Mayer or Ree-Hoover weights of particular graphs is very challenging in general and have been made for only certain specific families of graphs (see [4] [5] [6] [15] [16] [17] [18]). In the present paper we extend this list to include other graphs. We give new explicit formulas of the Ree-Hoover weight of these graphs in Section 3. Section 4 is devoted to the explicit computation of their Mayer weight. The following conventions are used in the present paper: Each graph g is identified with its set of edges. So that, { } i j is an edge in g between vertex i and vertex j. The number of edges in g is denoted ( ) e g . If e is an edge of g (i.e. e g ∈ ), \ g e denotes the graph obtained from g by removing the edge e. If and then expanding each weight In [1], we gived explicit linear relations expressing the Ree-Hoover weights in terms of the Mayer weights and vice versa: For a 2-connected graph b, we have So that the virial coefficient can be rewritten in the form for appropriate coefficients ( ) n a b called the star content of the graph b. The importance of (1.12) is due to the fact that for many graphs b. This greatly simplifies the computation of n β .
Using the definition of the Ree-Hoover weight, we have A. Kaouche

Hard-Core Continuum Gas in One Dimension
Consider n hard particles of diameter 1 on a line segment. The hard-core constraint translates into the interaction potential ϕ , with ( ) r ≥ , and the Mayer function f and the Ree-Hoover function f are given by (7). Hence, we can write the Mayer weight function (14) and the Ree-Hoover's weight function

Graph Homomorphisms
The method of graph homomorphisms was introduced in [6] for the calculation of the Mayer weight . Each simplice is represented by a diagram associated to the integral parts and the relative positions of the fractional parts of the coor- More specifically, to each real number x, they associate his fractional representation, which is a pair ( ) and it is shown in [1] that each such simplex is affine-equivalent to the standard simplex ( ) ( ) Using the fractional coordinates to represent the center of gravity , h β such that the condition (17) is satisfied. Then the volume of the polytope Proposition 1 can be used to compute the weight of some families of graphs, In a similar way we can adapt the above configurations to the context of the Ree-Hoover weight. , h β such that conditions (19) and (20) are satisfied. Then the volume of ( )

Ree-Hoover Weight of New Families of Graphs
In this section, we give other explicit formulas for the Ree-Hoover weight for infinite families of 2-connected graphs. First, we use Ehrhart polynomials to conjectured these formulas from numerical values. We use the techniques of graph homorphisms in order to prove these formulas. The weights of 2-connected graphs b are given in absolute value

Mayer Weight of New Families of Graphs
Here are some of our results concerning new explicit formulas for the Mayer weight of the previous infinite families of graphs. In this case, the computation of the Mayer weight is more difficult. Instead of using the method of graph homomorphisms, we use the following formula which is a consequence of (1.11) in the case of hard-core continuum gases in one dimension. Substituting \ n K g and \ n K k for b and d in (29) (5) and (6)-(10).

Conclusion
The links between statistical mechanics and combinatorics are more and more numerous as we have seen in this work. In this paper, after recalling the Mayer and Ree-Hoover theory, we presented in Section 2 the method of graph homomorphisms and we have mainly placed ourselves in the context of hard-core continuum gas in one dimension. From various tables that we constructed giving numerical values of Mayer and Ree-Hoover weights of all 2-connected graphs up to size 8, we conjectured explicit formulas for Mayer and Ree-Hoover weights of the family

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.