New Formulas for the Mayer and Ree-Hoover Weights of Infinite Families of Graphs ()
1. 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
, the Second Mayer weight
of a connected graph c, over the set
of vertices, is defined by (see [1] [2] [3] [4])
(1.1)
where
are variables in
representing the positions of n particles in V (
), the value
being arbitrarily fixed, and where
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
(1.2)
where k is a constant, T is the temperature and
is the density. Indeed, it can be shown that
where
denote the set of 2-connected graphs over
and
is the total sum of weights of 2-connected graphs over
. In order to compute this expansion numerically, Ree and Hoover [6] introduced a modified weight denoted by
, for 2-connected graphs b, which greatly simplifies the computations. It is defined by
(1.3)
where
. Using this new weight, Ree and Hoover [6] [7] [8] and later Clisby and McCoy [9] [10] have computed the virial coefficients
, for n up to 10, in dimensions
, in the case of the hard-core continuum gas, that is when the interaction is given by
(1.4)
where
denote de characteristic function. In this paper we study graph weights
and
in the context of the hard core continuum gas, defined by (1.4), in dimension
. The values
and
for all 2-connected 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
naturally associated with the graph c. An alternate useful tool, a decomposition of the polytope
into a certain number of
-dimensional simplices, of volume
is exploited in Section 2.2. This method was introduced in [4] and was adapted in [3] 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 difficult in general and have been made for only certain specific families of graphs (see [2] [3] [4] [12] [13] [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.
2. Mayer and Ree-Hoover Weights
2.1. 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
, with
, if
, and
, if
, and the Mayer function f and the Ree-Hoover function
are given by (1.4). Hence, we can write the Mayer weight function
of a connected graph c as
(2.1)
and the Ree-Hoover’s weight function
of a 2-connected graph c as
(2.2)
with
and where
is the number of edges of c.
Note that
, where
is the polytope defined by
where
. Similarly,
, where
is the union of polytopes defined by
2.2. Graph Homomorphisms
The method of graph homomorphisms was introduced in [4] for the exact computation of the Mayer weight
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
, the computation of
is reduced to the computation of the volume of the polytope
associated to b. In order to evaluate this volume, the polytope
is decomposed into
simplices which are all of volume
. This yields
. The simplices are encoded by a diagram associated to the integral parts and the relative positions of the fractional parts of the coordinates
of points
(see [3] [4] for more details).
Lemma 1. ([3]). Suppose that g is a graph over
and
are such that g does not contain the edge
but contains the edges
and
. In this case, any RH-configuration
(with
) satisfies either one of the following conditions:
1)
,
and
,
2)
,
and
.
2.3. 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)
(2.3)
2.3.1. The Mayer and Ree-Hoover Weight of the Graph
Let
denote the graph obtained by identifying two non adjacent vertices of the graph
with the extremities of a 2-star graph, where
is the cycle with 4 vertices and
denote the k-star graph with vertex set
and edge set
. See Figure 1.
Proposition 1. For
, we have
(2.4)
(2.5)
Proof. We can assume that the missing edges are
,
,
,
,
and
(see Figure 1).
According to Lemma 1 there are four possibilities for h:
−
and
and all other
, so that
and
must be a permutation of
.
−
and all other
, so that
and
must be a permutation of
.
−
and
and all other
, so that
and
must be a permutation of
.
−
and
and all other
, so that
and
must be a permutation of
.
In each case
can be extended in
ways, giving the possible relative positions of the
(see Figure 2). So, there are
RH- configurations
. Which concludes the proof of (2.6).
The over graphs of
whose Ree-Hoover weight is not zero and their multiplicities are given by
We conclude using Proposition (1) and Propositions (19)-(23) of [3].
2.3.2. The Mayer and Ree-Hoover Weight of the Graph
Let
denote the graph obtained by identifying one vertex, with
![]()
Figure 2. Fractional representation of a simplicial subpolytope of
.
degree two, of the graph
with a center of a j-star. See Figure 3 for an example.
Let us start with the case
.
Proposition 2. For
, we have
(2.6)
(2.7)
Proof. We can assume that the missing edges are
,
,
,
,
,
and
(see Figure 4).
According to Lemma 1 there are four possibilities for h:
−
and
and all other
, so that
and
must be a permutation of
and
and
.
−
and all other
, so that
and
must be a permutation of
and
and
.
−
and
and all other
, so that
and
must be a permutation of
and
and
.
−
and
and all other
, so that
and
must be a permutation of
and
and
.
In each case
can be extended in
ways, giving the possible relative positions of the
(see Figure 5). So, there are
RH- configurations
.
The over graphs of
whose Ree-Hoover weight is not zero are up to isomorphism of the form:
,
,
,
,
,
,
,
,
,
, and
. Their multiplicities are given by
We conclude using Propositions (1), (2) and Propositions (19)-(23) of [3].
In the general case we have:
Proposition 3. For
, we have, with the usual convention
if
,
![]()
Figure 5. Fractional representation of a simplicial subpolytope of
.
(2.8)
Proof. We can assume that the missing edges are
,
,
,
,
,
and
(see Figure 6, for the case of
).
According to Lemma 1 there are four possibilities for h:
−
and
and all other
, so that
and
must be a permutation of
and
must be a permutation of
and
.
−
and all other
, so that
and
must be a permutation of
and
must be a permutation of
and
.
−
and
and all other
, so that
and
must be a permutation of
and
must be a permutation of
and
.
−
and
and all other
, so that
and
must be a permutation of
and
must be a permutation of
and
.
In each case
can be extended in
ways, giving the possible relative positions of the
(see Figure 7, for the case of
). So, there are
RH-configurations
. Which concludes the proof of (2.8).
![]()
Figure 7. Fractional representation of a simplicial subpolytope of
.
The over graphs of
whose Ree-Hoover weight is not zero and their multiplicities are given by
We conclude using Propositions (1), (3) and Propositions (19)-(23) of [3].