New Formulas for the Mayer and Ree-Hoover Weights of Infinite Families of Graphs

The virial expansion, in statistical mechanics, makes use of the sums of the Mayer weight of all 2-connected graphs on n vertices. We study the Second Mayer weight ωM(c) and the Ree-Hoover weight ωRH(c) of a 2-connected graph c which arise from the hard-core continuum gas in one dimension. These weights are computed using signed volumes of convex polytopes naturally associated with the graph c. In the present work, we use the method of graph homomorphisms, to give new formulas of Mayer and Ree-Hoover weights for special infinite families of 2-connected graphs.


Introduction
Graph weights can be defined as functions on graphs taking scalar or polynomial values and which are invariant under isomorphism.In the context of a non-ideal gas in a vessel this expansion numerically, Ree and Hoover [6] introduced a modified weight denoted by ( ) RH w b , for 2-connected graphs b, which greatly simplifies the computations.It is defined by where ( )   [14]).In the present paper we extend this list to other graphs.We give new explicit formulas of the Mayer and Ree-Hoover weights for special infinite families of graphs in Section 2.3.

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 ( ) with 0 n x = and where ( ) e c is the number of edges of c.
Note that

Graph Homomorphisms
The method of graph homomorphisms was introduced in [4] for the exact computation of the Mayer weight ( ) M w b of a 2-connected graph b in the context of hard-core continuum gases in one dimension and was adapted in [3] to the context of Ree-Hoover weights.Since (see [3] [4] for more details).
Lemma 1. ( [3]).Suppose that g is a graph over [ ] n and , [ 1]  i j n ∈ − are such that g does not contain the edge { , } n i but contains the edges { , } i j and { , } n j .In this case, any RH-configuration ( , ) h β (with 1, ( ) satisfies either one of the following conditions: 1)

Mayer and Ree-Hoover Weights of Some Infinite Families of Graphs
Here are some of our results concerning new explicit formulas for the Ree-Hoover weight of certain infinite families of graphs.These were first conjectured from numerical values using Ehrhart polynomials.Their proofs use the techniques of graph homorphisms.We also give explicit formulas for the Mayer weight of the same infinite families of graphs.In order to do so, we use the following formula (see [3] for more details) .See Figure 1.
The over graphs of 4 2

\ ( )
n K C S ⋅⋅ whose Ree-Hoover weight is not zero and their multiplicities are given by We conclude using Proposition (1) and Propositions (19)-(23) of [3].


representing the positions of n particles in V ( V → ∞ ), the value 0 n x = being arbitrarily fixed, and where ( ) f f r = is real-valued function associated with the pairwise interaction potential of the particles, see [4] [5].The Mayer weight occur in the so-called virial expansion proposed by Kamerlingh Onnes in 1901 Engineering and Technologywhere k is a constant, T is the temperature and ρ is the density.Indeed, it can of weights of 2-connected graphs over [ ] n .In order to compute χ denote de characteristic function.In this paper we study graph weights the context of the hard core continuum gas, defined by (1.4), in dimension 1 d = .The values ( ) graphs c of size at most 8 are given in [1] [11].In Section 2.1, 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.An alternate useful tool, a decomposition of the polytope ( ) c  into a certain number of ( 1) n − -dimensional simplices, of volume

1 r
≥ , and the Mayer function f and the Ree-Hoover function f are given by (1.4).Hence, we can write the Mayer weight function ( ) and the Ree-Hoover's weight function ( ) RH w c of a 2-connected graph c as

1
reduced to the computation of the volume of the polytope ( ) b  associated to b.In order to evaluate this volume, the polytope ( ) b  is decomposed into ( ) b ν simplices which are all of volume 1/ ( 1)! n − .This yields Vol( are encoded by a diagram associated to the integral parts and the relative positions of the fractional parts of the coordinates 1 , ,
⋅ ⋅⋅ denote the graph obtained by identifying one vertex, with A. Kaouche DOI: 10.4236/wjet.2019.72019287 World Journal of Engineering and Technology

Figure 5 .
Figure 5. Fractional representation of a simplicial subpolytope of Figure 7, for the case of

Figure 7 .
Figure 7. Fractional representation of a simplicial subpolytope of , in the case of the hard-core continuum gas, that is when the interaction is given by