Other Formulas for the Ree-Hoover and Mayer Weights of Families of 2-Connected Graphs ()
1. 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
Definition 1. A simple graph g is formed of two sets: a non-empty finite set V, called the set of vertices of g, and a set E of pairs of vertices, called the set of edges of g. So we have
with
denotes all the parts of V with two elements. We often write
.
Definition 2. A subgraph h of a graph
is a graph of the form
, such that
and
.
Definition 3. An over graph g of a graph
is a graph of the form
, such that
and
.
In the present work it will be useful to identify a graph with all of its edges, that is to say
.
Definition 4. In a simple graph
, a chain c is a finite sequence of vertices,
, such that for all
,
. We write
.
Definition 5. A graph
is connected if
, 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:
there is a chain v to w in g. Let
the equivalence classes of ~ and let’s say, for
,
, the subgraph of g generated by
. These simple graphs
, 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 in a vessel
. In this case, the Second Mayer weight
of a connected graph c, over the set
of vertices, is defined by (see [1] [4] [5] [6] )
(1)
where
are variables in
representing the positions of n particles in V (
), the value
being arbitrarily fixed, and where
is a real-valued function associated with the pairwise interaction potential of the particles, see [6] [7] .
Let
be the set of connected graphs over
. The total sum of weights of connected graphs over
is denoted by
(2)
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] ):
Figure 1. A simple graph and its connected components.
(3)
where k is a constant, T is the temperature, and z is a variable called the fugacity or the activity of the system.
It is known that the weight
is multiplicative over 2-connected components so that in order to compute the weights
of the connected graphs
, it is sufficient to compute the weights
for 2-connected graphs
(
for blocks). The Mayer weight appear in the so-called virial expansion proposed by Kamerlingh Onnes in 1901
(4)
where
is the density. Indeed, it can be shown that
(5)
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 [8] introduced a modified weight denoted by
, for 2-connected graphs
which greatly simplifies the computations. It is defined by
(6)
where
. Using this new weight, Ree and Hoover [8] [9] [10] and later Clisby and McCoy [11] [12] [13] 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
(7)
where
denote the characteristic function (
, 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
. The values
and
for all 2-connected graphs c of size at most 8 are given in [1] [14] .
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
naturally associated with the graph c. A decomposition of the polytope
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,
means that
is an edge in g between vertex i and vertex j. The number of edges in g is denoted
. If e is an edge of g (i.e.
),
denotes the graph obtained from g by removing the edge e. If b and d are graphs,
means that b is a subgraph of d. The complete graph on the vertex set
is denoted by
. The complementary graph of a subgraph
is the graph
.
An important rewriting of the virial coefficients was performed by Ree and Hoover [8] [9] by introducing the function
(8)
and defining a new weight (denoted here by
) for 2-connected graphs b, by (9)
(9)
and then expanding each weight
by substituting
for pairs of vertices not connected by edges.
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
(10)
(11)
So that the virial coefficient can be rewritten in the form
(12)
for appropriate coefficients
called the star content of the graph b. The importance of (1.12) is due to the fact that
or
for many graphs b. This greatly simplifies the computation of
.
Using the definition of the Ree-Hoover weight, we have
(13)
2. 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 (7). Hence, we can write the Mayer weight function
of a connected graph c as
(14)
and the Ree-Hoover’s weight function
of a 2-connected graph c as
(15)
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
Graph Homomorphisms
The method of graph homomorphisms was introduced in [6] for the calculation of the Mayer weight
of a 2-connected graph b in the context of hard-core continuum gases in one dimension and was fited in [5] to the context of Ree-Hoover weights. Since
, the calculation of
is reduced to the calculation of the volume of the polytope
associated to b. In order to compute this volume, the polytope
is decomposed into
simplices which are all of volume
and we will have
. Each simplice is represented by a diagram associated to the integral parts and the relative positions of the fractional parts of the coordinates
of points
.
More specifically, to each real number x, they associate his fractional representation, which is a pair
, where
is the integral part of x and
is the (positive) fractional part of x, so that
. Then, for
, the condition
translates into “assuming
, then
or
”. It mean that the slope of the line segment between the points
and
in the plane should be either null or negative. Let b a 2-connected graph with vertex set
, and let
be a point in the polytope
. Let’s write
for the fractional representation of the coordinate
of X. For
, it will be convenient to use the special representation
and
. Remarque that the volume of
is unchanged by removing all hyperplanes
, for
. in consequence, we can assume that all the fractional parts
are distinct. We get a subpolytope of
by fixing the “heights”
as well as the relative positions (total order) of the fractional parts
. Let
denote the height function
and
be the permutation of
for which
gives the rank of
in this total order with
. Explicitly, each simplex
can be written as
(16)
and it is shown in [1] that each such simplex is affine-equivalent to the standard simplex
of volume
.
Note that the simplices (16) are disjoint and each such simplex can be characterized by its centre of gravity
Note also that when there are no restrictions on h and
, the union of the closed simplices
coincides with the whole configurations space
.
Using the fractional coordinates to represent the center of gravity
of the simplex
, and drawing a line segment form
and
for each edge
of the graph b, we get a configuration in the plane which is an homomorphic image of b which represents the subpolytope
. The above content is summarized in to the form of a proposition:
Proposition 1. ( [6] ). Let b be a 2-connected graph with vertex set
and consider a function
and a bijection
satisfying
. Then the simplex
corresponding to the pair
is contained in the polytope
if and only if the following condition is satisfied:
for any edge
of b,
implies
or
. (17)
Corollary 1. ( [6] ). Let b be a 2-connected graph and let
be the number of pairs
such that the condition (17) is satisfied. Then the volume of the polytope
is given by
(18)
Proposition 1 can be used to compute the weight of some families of graphs, since
.
In a similar way we can adapt the above configurations to the context of the Ree-Hoover weight.
Proposition 2. ( [5] ). Let b be a 2-connected graph with vertex set
and consider a function
and a bijection
satisfying
. Then the simplex
corresponding to the pair
is contained in the polytope
if and only if the following conditions are satisfied:
for any edge
of b,
implies
or
. (19)
for any edge
of
,
implies
or
. (20)
Proposition 3. ( [5] ). Let b be a 2-connected graph and let
be the number of pairs
such that conditions (19) and (20) are satisfied. Then the volume of
is given by
(21)
3. 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
, the sign being always equal to
.
Lemma 1. ( [5] ). 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
.
3.1. The Ree-Hoover Weight of the Graph
Let
denote the graph obtained by identifying one vertex, with degree three, of the graph
with a center of a k-star. See Figure 3 for an example.
Let us start with the simple case
.
Proposition 4. For
, we have
(22)
Proof. We can assume that the missing edges are
,
,
,
,
,
and
(see Figure 4).
According to Lemma 1 there are two possibilities for h:
·
and
and all other
, so that
and
and
must be a permutation of
.
·
and all other
, so that
and
and
must be a permutation of
.
In each case
can be extended in
ways, giving the possible relative positions of the
(see Figure 5). So, there are
RH-configurations
.
Figure 3. The graph
.
Figure 4. The graph
.
Figure 5. Fractional representation of a simplicial subpolytope of
.
3.2. The Ree-Hoover Weight of the Graph
In the general case we have:
Proposition 5. For
,
, we have
(23)
Proof. We can assume that the missing edges are
,
,
,
,
,
and
,
,
,
(see Figure 6, for the case of
).
According to Lemma 1 there are two possibilities for h:
·
and
and all other
, so that
must be a permutation of
and
and
must be a permutation of
.
·
and all other
, so that
must be a permutation of
and
and
must be a permutation of
.
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
.
We need to use Propositions (6)-(10) to prove Mayer’s weight formulas that will be presented in section 4.
Proposition 6. ( [5] ). For
, we have
(24)
where
is the unoriented cycle with 4 vertices.
Proposition 7. ( [5] ). For
,
, we have
(25)
where
denote the k-star graph with vertex set
and edge set
, (see Figure 8, for the case of
).
Proposition 8. ( [5] ). For
,
, we have
(26)
where
denote the graph obtained by joining with a new edge the centers of a j-star and of a k-star. See Figure 9 for an example.
Proposition 9. ( [5] ). For
,
, we have
(27)
where
denote the graph obtained by identifying one vertex of the graph
with the center of a k-star. See Figure 10 for an example.
Proposition 10. ( [18] ). For
, we have
(28)
where
denote the graph obtained by identifying two non adjacent vertices of the graph
(the unoriented cycle with 4 vertices) with the extremities of a 2-star (see Figure 11).
Figure 6. The graph
.
Figure 7. Fractional representation of a simplicial subpolytope of
.
4. 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
(29)
which is a consequence of (1.11) in the case of hard-core continuum gases in one dimension. Substituting
and
for b and d in (29), we have
(30)
where
denotes the unlabelled graph corresponding to g,
runs through the unlabelled subgraphs of
and
is the number of ways of obtaining
by removing some edges in
. We obtain these multiplicities
by combinatorial arguments.
4.1. The Mayer Weight of the Graph
Proposition 11. For
, we have
(31)
Proof. 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 (5) and (6)-(10).
4.2. The Mayer Weight of the Graph
Proposition 12. For
,
,
, we have, with the usual convention
if
,
Proof. 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 (5) and (6)-(10).
5. 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
,
, and more generally for the family
,
,
. These formulas have been proved in Section 3 by the method of graph homomorphisms for the Ree-Hoover weight and by the linear relations between the two weights for Mayer’s weight in Section 4. A similar work was done by the author, see [18] , for families of graphs
,
and
,
, and more generally for families
,
,
. These developments pave the way for several future research prospects. For example, the extension of the exact calculation of Mayer’s weight and Ree-Hoover’s weight for families of graphs
,
and
,
and
,
and more generally for families
,
,
and
,
,
, with
denote the graph obtained by joining with an edge of the graph
the centers of a j-star and k-star.