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

Journal of Applied Mathematics and Physics
Vol.07 No.08(2019), Article ID:94424,14 pages
10.4236/jamp.2019.78123

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

Amel Kaouche

Université de Moncton, Campus d’Edmundston, Edmundston, Canada    Received: June 28, 2019; Accepted: August 16, 2019; Published: August 19, 2019

ABSTRACT

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.

Keywords:

Combinatorial, Mayer Weight, Statistical Mechanics, Ree-Hoover Weight, Graph Invariants, Virial Expansion 1. Introduction

Before discussing our subject, we first present some preliminary notions on the theory of graphs drawn from among others    .

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 $E\subseteq {\mathcal{P}}_{2}\left(V\right)$ with ${\mathcal{P}}_{2}\left(V\right)$ denotes all the parts of V with two elements. We often write $g=\left(V,E\right)$ .

Definition 2. A subgraph h of a graph $g=\left(V,E\right)$ is a graph of the form $h=\left({V}_{0},{E}_{0}\right)$ , such that ${V}_{0}\subseteq V$ and ${E}_{0}={\mathcal{P}}_{2}\left({V}_{0}\right)\cap E$ .

Definition 3. An over graph g of a graph $h=\left(V,E\right)$ is a graph of the form $g=\left({V}_{1},{E}_{1}\right)$ , such that $V\subseteq {V}_{1}$ and $E={\mathcal{P}}_{2}\left(V\right)\cap {E}_{1}$ .

In the present work it will be useful to identify a graph with all of its edges, that is to say $g\subseteq {\mathcal{P}}_{2}\left(V\right)$ .

Definition 4. In a simple graph $g=\left(V,E\right)$ , a chain c is a finite sequence of vertices, ${v}_{0},{v}_{1},\cdots ,{v}_{m}$ , such that for all $0\le i , $\left\{{v}_{i},{v}_{i+1}\right\}\in E$ . We write $c=\left[{v}_{0},{v}_{1},\cdots ,{v}_{m}\right]$ .

Definition 5. A graph $g=\left(V,E\right)$ is connected if $\forall v,w\in V$ , 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 $g=\left(V,E\right)$ , we define the relation of equivalence: $v~w$ $⇔$ there is a chain v to w in g. Let ${V}_{1},{V}_{2},\cdots ,{V}_{k}$ the equivalence classes of ~ and let’s say, for $1\le i\le k$ , ${g}_{i}={g}_{{V}_{i}}$ , the subgraph of g generated by ${V}_{i}$ . These simple graphs ${g}_{i}$ , 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 $V\subseteq {ℝ}^{d}$ . In this case, the Second Mayer weight ${w}_{M}\left(c\right)$ of a connected graph c, over the set $\left[n\right]=\left\{1,2,\cdots ,n\right\}$ of vertices, is defined by (see     )

${w}_{M}\left(c\right)={\int }_{{\left({ℝ}^{d}\right)}^{n-1}}\underset{\left\{i,j\right\}\in c}{\prod }f\left(‖{x}_{i}-{x}_{j}‖\right)\text{d}{x}_{1}\cdots \text{d}{x}_{n-1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{n}=0,$ (1)

where ${x}_{1},\cdots ,{x}_{n}$ are variables in ${ℝ}^{d}$ representing the positions of n particles in V ( $V\to \infty$ ), the value ${x}_{n}=0$ being arbitrarily fixed, and where $f=f\left(r\right)$ is a real-valued function associated with the pairwise interaction potential of the particles, see   .

Let $\mathcal{C}\left[n\right]$ be the set of connected graphs over $\left[n\right]$ . The total sum of weights of connected graphs over $\left[n\right]$ is denoted by

${|\mathcal{C}\left[n\right]|}_{{w}_{M}}=\underset{c\in \mathcal{C}\left[n\right]}{\sum }{w}_{M}\left(c\right).$ (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  ):

Figure 1. A simple graph and its connected components.

Figure 2. A 2-connected graph.

$\frac{P}{kT}={\mathcal{C}}_{{w}_{M}}\left(z\right)=\underset{n\ge 1}{\sum }{|\mathcal{C}\left[n\right]|}_{{w}_{M}}\frac{{z}^{n}}{n!},$ (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 ${w}_{M}$ is multiplicative over 2-connected components so that in order to compute the weights ${w}_{M}\left(c\right)$ of the connected graphs $c\in \mathcal{C}\left[n\right]$ , it is sufficient to compute the weights ${w}_{M}\left(b\right)$ for 2-connected graphs $b\in \mathcal{B}\left[n\right]$ ( $\mathcal{B}$ for blocks). The Mayer weight appear in the so-called virial expansion proposed by Kamerlingh Onnes in 1901

$\frac{P}{kT}=\rho +{\beta }_{2}{\rho }^{2}+{\beta }_{3}{\rho }^{3}+\cdots ,$ (4)

where $\rho$ is the density. Indeed, it can be shown that

${\beta }_{n}=\frac{1-n}{n!}{|\mathcal{B}\left[n\right]|}_{{w}_{M}},$ (5)

where $\mathcal{B}\left[n\right]$ denote the set of 2-connected graphs over $\left[n\right]$ and ${|\mathcal{B}\left[n\right]|}_{{w}_{M}}$ is the total sum of weights of 2-connected graphs over $\left[n\right].$ In order to compute this expansion numerically, Ree and Hoover  introduced a modified weight denoted by ${w}_{RH}\left(b\right)$ , for 2-connected graphs $b,$ which greatly simplifies the computations. It is defined by

${w}_{RH}\left(b\right)={\int }_{{\left({ℝ}^{d}\right)}^{n-1}}\underset{\left\{i,j\right\}\in b}{\prod }f\left(‖{x}_{i}-{x}_{j}‖\right)\underset{\left\{i,j\right\}\notin b}{\prod }\stackrel{¯}{f}\left(‖{x}_{i}-{x}_{j}‖\right)\text{d}{x}_{1}\cdots \text{d}{x}_{n-1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{n}=0,$ (6)

where $\stackrel{¯}{f}\left(r\right)=1+f\left(r\right)$ . Using this new weight, Ree and Hoover    and later Clisby and McCoy    have computed the virial coefficients ${\beta }_{n}$ , for n up to 10, in dimensions $d\le 8$ , in the case of the hard-core continuum gas, that is when the interaction is given by

$f\left(r\right)=-\chi \left(r<1\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{¯}{f}\left(r\right)=\chi \left(r\ge 1\right),$ (7)

where $\chi$ denote the characteristic function ( $\chi \left(P\right)=1$ , 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 $d=1$ . The values ${w}_{M}\left(c\right)$ and ${w}_{RH}\left(c\right)$ for all 2-connected graphs c of size at most 8 are given in   .

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 $\mathcal{P}\left(c\right)$ naturally associated with the graph c. A decomposition of the polytope $\mathcal{P}\left(c\right)$ into a certain number of simplices is utilised. This method was introduced in  and was adapted in   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        ). 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, $\left\{i,j\right\}\in g$ means that $\left\{i,j\right\}$ is an edge in g between vertex i and vertex j. The number of edges in g is denoted $e\left(g\right)$ . If e is an edge of g (i.e. $e\in g$ ), $g\e$ denotes the graph obtained from g by removing the edge e. If b and d are graphs, $b\subseteq d$ means that b is a subgraph of d. The complete graph on the vertex set $\left[n\right]=\left\{1,2,\cdots ,n\right\}$ is denoted by ${K}_{n}$ . The complementary graph of a subgraph $g\subseteq {K}_{n}$ is the graph $\stackrel{¯}{g}={K}_{n}\g$ .

An important rewriting of the virial coefficients was performed by Ree and Hoover   by introducing the function

$\stackrel{¯}{f}\left(r\right)=1+f\left(r\right)$ (8)

and defining a new weight (denoted here by ${w}_{RH}\left(b\right)$ ) for 2-connected graphs b, by (9)

${w}_{RH}\left(b\right)={\int }_{{\left({ℝ}^{d}\right)}^{n-1}}\prod _{\left\{i,j\right\}\in b}f\left(‖{x}_{i}-{x}_{j}‖\right)\prod _{\left\{i,j\right\}\notin b}\text{ }\overline{f}\left(‖{x}_{i}-{x}_{j}‖\right)\text{d}{x}_{1}\cdots \text{d}{x}_{n-1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{n}=0,$ (9)

and then expanding each weight ${w}_{M}\left(b\right)$ by substituting $1=\stackrel{¯}{f}-f$ for pairs of vertices not connected by edges.

In  , 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

${w}_{RH}\left(b\right)=\underset{b\subseteq d\subseteq {K}_{n}}{\sum }{w}_{M}\left(d\right),$ (10)

${w}_{M}\left(b\right)=\underset{b\subseteq d\subseteq {K}_{n}}{\sum }{\left(-1\right)}^{e\left(d\right)-e\left(b\right)}{w}_{RH}\left(d\right).$ (11)

So that the virial coefficient can be rewritten in the form

${\beta }_{n}=\frac{1-n}{n!}\underset{b\in \mathcal{B}\left[n\right]}{\sum }{a}_{n}\left(b\right){w}_{RH}\left(b\right),$ (12)

for appropriate coefficients ${a}_{n}\left(b\right)$ called the star content of the graph b. The importance of (1.12) is due to the fact that ${a}_{n}\left(b\right)=0$ or ${w}_{RH}\left(b\right)=0$ for many graphs b. This greatly simplifies the computation of ${\beta }_{n}$ .

Using the definition of the Ree-Hoover weight, we have

${w}_{RH}\left({K}_{n}\right)={w}_{M}\left({K}_{n}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}n\ge 2.$ (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 $\phi$ , with $\phi \left(r\right)=\infty$ , if $r<1$ , and $\phi \left(r\right)=0$ , if $r\ge 1$ , and the Mayer function f and the Ree-Hoover function $\stackrel{¯}{f}$ are given by (7). Hence, we can write the Mayer weight function ${w}_{M}\left(c\right)$ of a connected graph c as

${w}_{M}\left(c\right)={\left(-1\right)}^{e\left(c\right)}{\int }_{{ℝ}^{n-1}}\underset{\left\{i,j\right\}\in c}{\prod }\chi \left(|{x}_{i}-{x}_{j}|<1\right)\text{d}{x}_{1}\cdots \text{d}{x}_{n-1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{n}=0,$ (14)

and the Ree-Hoover’s weight function ${w}_{RH}\left(c\right)$ of a 2-connected graph c as

${w}_{RH}\left(c\right)={\left(-1\right)}^{e\left(c\right)}{\int }_{{ℝ}^{n-1}}\prod _{\left\{i,j\right\}\in c}\chi \left(|{x}_{i}-{x}_{j}|<1\right)\prod _{\left\{i,j\right\}\notin c}\chi \left(|{x}_{i}-{x}_{j}|>1\right)\text{d}{x}_{1}\cdots \text{d}{x}_{n-1},$ (15)

with ${x}_{n}=0$ and where $e\left(c\right)$ is the number of edges of c. Note that ${w}_{M}\left(c\right)={\left(-1\right)}^{e\left(c\right)}\text{Vol}\left(\mathcal{P}\left(c\right)\right)$ , where $\mathcal{P}\left(c\right)$ is the polytope defined by

$\mathcal{P}\left(c\right)=\left\{X\in {ℝ}^{n}|{x}_{n}=0,|{x}_{i}-{x}_{j}|<1\text{\hspace{0.17em}}\text{ }\forall \left\{i,j\right\}\in c\right\}\subseteq {ℝ}^{n-1}×\left\{0\right\}\subseteq {ℝ}^{n},$

where $X=\left({x}_{1},\cdots ,{x}_{n}\right)$ . Similarly, ${w}_{RH}\left(c\right)={\left(-1\right)}^{e\left(c\right)}\text{Vol}\left({\mathcal{P}}_{RH}\left(c\right)\right)$ , where ${\mathcal{P}}_{RH}\left(c\right)$ is the union of polytopes defined by

${\mathcal{P}}_{RH}\left(c\right)=\left\{X\in {ℝ}^{n}|{x}_{n}=0,|{x}_{i}-{x}_{j}|<1\text{\hspace{0.17em}}\text{ }\forall \left\{i,j\right\}\in c,|{x}_{i}-{x}_{j}|>1\text{\hspace{0.17em}}\text{ }\forall \left\{i,j\right\}\in \stackrel{¯}{c}\right\}.$

Graph Homomorphisms

The method of graph homomorphisms was introduced in  for the calculation of the Mayer weight ${w}_{M}\left(b\right)$ of a 2-connected graph b in the context of hard-core continuum gases in one dimension and was fited in  to the context of Ree-Hoover weights. Since ${w}_{M}\left(b\right)={\left(-1\right)}^{e\left(b\right)}\text{Vol}\left(\mathcal{P}\left(b\right)\right)$ , the calculation of ${w}_{M}\left(b\right)$ is reduced to the calculation of the volume of the polytope $\mathcal{P}\left(b\right)$ associated to b. In order to compute this volume, the polytope $\mathcal{P}\left(b\right)$ is decomposed into $\nu \left(b\right)$ simplices which are all of volume $1/\left(n-1\right)!$ and we will have $\text{Vol}\left(\mathcal{P}\left(b\right)\right)=\nu \left(b\right)/\left(n-1\right)!$ . Each simplice is represented by a diagram associated to the integral parts and the relative positions of the fractional parts of the coordinates ${x}_{1},\cdots ,{x}_{n}$ of points $X\in \mathcal{P}\left(b\right)$ .

More specifically, to each real number x, they associate his fractional representation, which is a pair $\left({\xi }_{x},{h}_{x}\right)$ , where ${h}_{x}=⌊x⌋$ is the integral part of x and ${\xi }_{x}=x-{h}_{x}$ is the (positive) fractional part of x, so that $x={\xi }_{x}+{h}_{x}$ . Then, for $x\ne y$ , the condition $|x-y|<1$ translates into “assuming ${\xi }_{x}<{\xi }_{y}$ , then ${h}_{x}={h}_{y}$ or ${h}_{x}={h}_{y}+1$ ”. It mean that the slope of the line segment between the points $\left({\xi }_{x},{h}_{x}\right)$ and $\left({\xi }_{y},{h}_{y}\right)$ in the plane should be either null or negative. Let b a 2-connected graph with vertex set $\left[n\right]$ , and let $X=\left({x}_{1},\cdots ,{x}_{n}\right)$ be a point in the polytope $\mathcal{P}\left(b\right)$ . Let’s write $\left({\xi }_{i},{h}_{i}\right)$ for the fractional representation of the coordinate ${x}_{i}$ of X. For ${x}_{n}=0$ , it will be convenient to use the special representation ${\xi }_{n}=1.0$ and ${h}_{n}=-1$ . Remarque that the volume of $\mathcal{P}\left(b\right)$ is unchanged by removing all hyperplanes $\left\{{x}_{i}-{x}_{j}=k\right\}$ , for $k\in ℤ$ . in consequence, we can assume that all the fractional parts ${\xi }_{i}$ are distinct. We get a subpolytope of $\mathcal{P}\left(b\right)$ by fixing the “heights” ${h}_{1},{h}_{2},\cdots ,{h}_{n}$ as well as the relative positions (total order) of the fractional parts ${\xi }_{1},{\xi }_{2},\cdots ,{\xi }_{n}$ . Let $h:V\to ℤ$ denote the height function $i↦{h}_{i}$ and $\beta :V\to \left[n\right]$ be the permutation of $\left[n\right]$ for which $\beta \left(i\right)$ gives the rank of ${\xi }_{i}$ in this total order with $\beta \left(n\right)=n$ . Explicitly, each simplex $\mathcal{P}\left(h,\beta \right)$ can be written as

$\mathcal{P}\left(h,\beta \right)=\left\{\left({h}_{1}+{\xi }_{1},\cdots ,{h}_{n-1}+{\xi }_{n-1},0\right)|0<{\xi }_{{\beta }^{-1}\left(1\right)}<\cdots <{\xi }_{{\beta }^{-1}\left(n-1\right)}<1\right\}$ (16)

and it is shown in  that each such simplex is affine-equivalent to the standard simplex

$\mathcal{P}\left(0,\text{id}\right)=\left\{\left({\xi }_{1},{\xi }_{2},\cdots ,{\xi }_{n-1},0\right)|0<{\xi }_{1}<{\xi }_{2}<\cdots <{\xi }_{n-1}<1\right\}$

of volume $1/\left(n-1\right)!$ .

Note that the simplices (16) are disjoint and each such simplex can be characterized by its centre of gravity

${X}_{h,\beta }=\left({h}_{1}+\frac{\beta \left(1\right)}{n},{h}_{2}+\frac{\beta \left(2\right)}{n},\cdots ,{h}_{n-1}+\frac{\beta \left(n-1\right)}{n},0\right).$

Note also that when there are no restrictions on h and $\beta$ , the union of the closed simplices $\stackrel{¯}{\mathcal{P}\left(h,\beta \right)}$ coincides with the whole configurations space ${ℝ}^{n-1}×\left\{0\right\}$ .

Using the fractional coordinates to represent the center of gravity ${X}_{h,\beta }$ of the simplex $\mathcal{P}\left(h,\beta \right)$ , and drawing a line segment form ${x}_{i}=\left({h}_{i},{\xi }_{i}\right)$ and ${x}_{j}=\left({h}_{j},{\xi }_{j}\right)$ for each edge $\left\{i,j\right\}$ of the graph b, we get a configuration in the plane which is an homomorphic image of b which represents the subpolytope $\mathcal{P}\left(h,\beta \right)$ . The above content is summarized in to the form of a proposition:

Proposition 1. (  ). Let b be a 2-connected graph with vertex set $V=\left[n\right]$ and consider a function $h:V\to ℤ$ and a bijection $\beta :V\to \left[n\right]$ satisfying $\beta \left(n\right)=n$ . Then the simplex $\mathcal{P}\left(h,\beta \right)$ corresponding to the pair $\left(h,\beta \right)$ is contained in the polytope $\mathcal{P}\left(\beta \right)$ if and only if the following condition is satisfied:

for any edge $\left\{i,j\right\}$ of b, $\beta \left(i\right)<\beta \left(j\right)$ implies ${h}_{i}={h}_{j}$ or ${h}_{i}={h}_{j}+1$ . (17)

Corollary 1. (  ). Let b be a 2-connected graph and let $\nu \left(b\right)$ be the number of pairs $\left(h,\beta \right)$ such that the condition (17) is satisfied. Then the volume of the polytope $\mathcal{P}\left(b\right)$ is given by

$\text{Vol}\left(\mathcal{P}\left(b\right)\right)=\nu \left(b\right)/\left(n-1\right)!.$ (18)

Proposition 1 can be used to compute the weight of some families of graphs, since ${w}_{M}\left(b\right)={\left(-1\right)}^{e\left(b\right)}\text{Vol}\left(\mathcal{P}\left(b\right)\right)$ .

In a similar way we can adapt the above configurations to the context of the Ree-Hoover weight.

Proposition 2. (  ). Let b be a 2-connected graph with vertex set $V=\left[n\right]$ and consider a function $h:V\to ℤ$ and a bijection $\beta :V\to \left[n\right]$ satisfying $\beta \left(n\right)=n$ . Then the simplex $\mathcal{P}\left(h,\beta \right)$ corresponding to the pair $\left(h,\beta \right)$ is contained in the polytope ${\mathcal{P}}_{RH}\left(b\right)$ if and only if the following conditions are satisfied:

for any edge $\left\{i,j\right\}$ of b, $\beta \left(i\right)<\beta \left(j\right)$ implies ${h}_{i}={h}_{j}$ or ${h}_{i}={h}_{j}+1$ . (19)

for any edge $\left\{i,j\right\}$ of $\stackrel{¯}{b}$ , $\beta \left(i\right)<\beta \left(j\right)$ implies ${h}_{i}\le {h}_{j}-1$ or ${h}_{i}\ge {h}_{j}+2$ . (20)

Proposition 3. (  ). Let b be a 2-connected graph and let ${\nu }_{RH}\left(b\right)$ be the number of pairs $\left(h,\beta \right)$ such that conditions (19) and (20) are satisfied. Then the volume of ${\mathcal{P}}_{RH}\left(b\right)$ is given by

$\text{Vol}\left({\mathcal{P}}_{RH}\left(b\right)\right)={\nu }_{RH}\left(b\right)/\left(n-1\right)!.$ (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 $|w\left(b\right)|$ , the sign being always equal to ${\left(-1\right)}^{e\left(b\right)}$ .

Lemma 1. (  ). Suppose that g is a graph over $\left[n\right]$ and $i,j\in \left[n-1\right]$ are such that g does not contain the edge $\left\{n,i\right\}$ but contains the edges $\left\{i,j\right\}$ and $\left\{n,j\right\}$ . In this case, any RH-configuration $\left(h,\beta \right)$ (with ${h}_{n}=-1$ , $\beta \left(n\right)=n$ ) satisfies either one of the following conditions:

1) ${h}_{i}=1$ , ${h}_{j}=0$ and $\beta \left(i\right)<\beta \left(j\right)$ ,

2) ${h}_{i}=-2$ , ${h}_{j}=-1$ and $\beta \left(i\right)>\beta \left(j\right)$ .

3.1. The Ree-Hoover Weight of the Graph ${K}_{n}\\left(\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}\right)$

Let $\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{k}$ denote the graph obtained by identifying one vertex, with degree three, of the graph $\left({C}_{4}\cdot \cdot {S}_{2}\right)$ with a center of a k-star. See Figure 3 for an example.

Let us start with the simple case $\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}$ .

Proposition 4. For $n\ge 7$ , we have

$|{w}_{RH}\left({K}_{n}\\left(\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}\right)\right)|=\frac{12}{\left(n-1\right)\left(n-2\right)\left(n-3\right)\left(n-4\right)\left(n-5\right)}\cdot$ (22)

Proof. We can assume that the missing edges are $\left\{1,n\right\}$ , $\left\{2,n\right\}$ , $\left\{4,n\right\}$ , $\left\{n,5\right\}$ , $\left\{1,3\right\}$ , $\left\{3,4\right\}$ and $\left\{2,3\right\}$ (see Figure 4).

According to Lemma 1 there are two possibilities for h:

· ${h}_{1}={h}_{2}={h}_{4}={h}_{5}=1$ and ${h}_{n}=-1$ and all other ${h}_{i}=0$ , so that $\beta \left(5\right)=1$ and $\beta \left(3\right)=2$ and $\left(\beta \left(1\right),\beta \left(2\right),\beta \left(4\right)\right)$ must be a permutation of $\left\{3,4,5\right\}$ .

· ${h}_{1}={h}_{2}={h}_{4}={h}_{5}=-2$ and all other ${h}_{i}=-1$ , so that $\beta \left(5\right)=n-1$ and $\beta \left(3\right)=n-2$ and $\left(\beta \left(1\right),\beta \left(2\right),\beta \left(4\right)\right)$ must be a permutation of $\left\{n-3,n-4,n-5\right\}$ .

In each case $\beta$ can be extended in $\left(n-6\right)!$ ways, giving the possible relative positions of the $\left(n-6\right)$ ${x}_{i}$ (see Figure 5). So, there are $2\cdot 3!\left(n-6\right)!$ RH-configurations $\left(h,\beta \right)$ .

Figure 3. The graph $\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{4}$ .

Figure 4. The graph $\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}$ .

Figure 5. Fractional representation of a simplicial subpolytope of ${\mathcal{P}}_{RH}\left({K}_{n}\\left(\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}\right)\right)$ .

3.2. The Ree-Hoover Weight of the Graph ${K}_{n}\\left(\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{k}\right)$

In the general case we have:

Proposition 5. For $k\ge 1$ , $n\ge k+6$ , we have

$|{w}_{RH}\left({K}_{n}\\left(\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{k}\right)\right)|=\frac{12k!}{\left(n-1\right)\left(n-2\right)\cdots \left(n-k-4\right)}.$ (23)

Proof. We can assume that the missing edges are $\left\{1,n\right\}$ , $\left\{2,n\right\}$ , $\left\{4,n\right\}$ , $\left\{3,4\right\}$ , $\left\{2,3\right\}$ , $\left\{1,3\right\}$ and $\left\{5,n\right\}$ , $\left\{6,n\right\}$ , $\cdots$ , $\left\{k+4,n\right\}$ (see Figure 6, for the case of $\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{2}$ ).

According to Lemma 1 there are two possibilities for h:

· ${h}_{1}={h}_{2}={h}_{4}={h}_{5}=\cdots ={h}_{k+4}=1$ and ${h}_{n}=-1$ and all other ${h}_{i}=0$ , so that $\left(\beta \left(5\right),\beta \left(6\right),\beta \left(7\right),\cdots ,\beta \left(k+4\right)\right)$ must be a permutation of $\left\{1,2,3,\cdots ,k\right\}$ and $\beta \left(3\right)=k+1$ and $\left(\beta \left(1\right),\beta \left(2\right),\beta \left(4\right)\right)$ must be a permutation of $\left\{k+2,k+3,k+4\right\}$ .

· ${h}_{1}={h}_{2}={h}_{4}={h}_{5}=\cdots ={h}_{k+4}=-2$ and all other ${h}_{i}=-1$ , so that

$\left(\beta \left(5\right),\beta \left(6\right),\beta \left(7\right),\cdots ,\beta \left(k+4\right)\right)$ must be a permutation of $\left\{n-1,n-2,n-3,\cdots ,n-k\right\}$ and $\beta \left(3\right)=n-k-1$ and $\left(\beta \left(1\right),\beta \left(2\right),\beta \left(4\right)\right)$ must be a permutation of $\left\{n-k-2,n-k-3,n-k-4\right\}$ .

In each case $\beta$ can be extended in $\left(n-k-5\right)!$ ways, giving the possible relative positions of the $\left(n-k-5\right)$ ${x}_{i}$ (see Figure 7, for the case of ${S}_{2}\cdot {C}_{4}\cdot \cdot {S}_{2}$ ). So, there are $2\cdot 3!k!\left(n-k-5\right)!$ RH-configurations $\left(h,\beta \right)$ .

We need to use Propositions (6)-(10) to prove Mayer’s weight formulas that will be presented in section 4.

Proposition 6. (  ). For $n\ge 6$ , we have

$|{w}_{RH}\left({K}_{n}\{C}_{4}\right)|=\frac{8}{\left(n-1\right)\left(n-2\right)\left(n-3\right)},$ (24)

where ${C}_{4}$ is the unoriented cycle with 4 vertices.

Proposition 7. (  ). For $k\ge 1$ , $n\ge k+3$ , we have

$|{w}_{RH}\left({K}_{n}\{S}_{k}\right)|=\frac{2k!}{\left(n-1\right)\left(n-2\right)\cdots \left(n-k\right)},$ (25)

where ${S}_{k}$ denote the k-star graph with vertex set $\left[k+1\right]$ and edge set $\left\{\left\{1,2\right\},\left\{1,3\right\},\cdots ,\left\{1,k+1\right\}\right\}$ , (see Figure 8, for the case of ${S}_{3}$ ).

Proposition 8. (  ). For $j\ge k\ge 1$ , $n\ge k+j+3$ , we have

$|{w}_{RH}\left({K}_{n}\\left({S}_{j}-{S}_{k}\right)\right)|=\frac{2k!j!}{\left(n-1\right)\left(n-2\right)\cdots \left(n-\left(k+j+1\right)\right)},$ (26)

where ${S}_{j}-{S}_{k}$ 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. (  ). For $k\ge 1$ , $n\ge k+5$ , we have

$|{w}_{RH}\left({K}_{n}\\left({C}_{4}\cdot {S}_{k}\right)\right)|=\frac{4k!}{\left(n-1\right)\left(n-2\right)\cdots \left(n-\left(k+3\right)\right)},$ (27)

where ${C}_{4}\cdot {S}_{k}$ denote the graph obtained by identifying one vertex of the graph ${C}_{4}$ with the center of a k-star. See Figure 10 for an example.

Proposition 10. (  ). For $n\ge 7$ , we have

$|{w}_{RH}\left({K}_{n}\\left({C}_{4}\cdot \cdot {S}_{2}\right)\right)|=\frac{24}{\left(n-1\right)\left(n-2\right)\left(n-3\right)\left(n-4\right)},$ (28)

where $\left({C}_{4}\cdot \cdot {S}_{2}\right)$ denote the graph obtained by identifying two non adjacent vertices of the graph ${C}_{4}$ (the unoriented cycle with 4 vertices) with the extremities of a 2-star (see Figure 11).

Figure 6. The graph $\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{2}$ .

Figure 7. Fractional representation of a simplicial subpolytope of ${\mathcal{P}}_{RH}\left({K}_{n}\\left(\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{2}\right)\right)$ .

Figure 8. The graph S3.

Figure 9. The graph ${S}_{3}-{S}_{4}$ .

Figure 10. The graph ${C}_{4}\cdot {S}_{4}$ .

Figure 11. The graph ${C}_{4}\cdot \cdot {S}_{2}$ .

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

$|{w}_{M}\left(b\right)|=\underset{b\subseteq d\subseteq {K}_{n}}{\sum }|{w}_{RH}\left(d\right)|$ (29)

which is a consequence of (1.11) in the case of hard-core continuum gases in one dimension. Substituting ${K}_{n}\g$ and ${K}_{n}\k$ for b and d in (29), we have

$|{w}_{M}\left({K}_{n}\g\right)|=\underset{k\subseteq g}{\sum }|{w}_{RH}\left({K}_{n}\k\right)|=\underset{\stackrel{˜}{h}\subseteq \stackrel{˜}{g}}{\sum }\text{ }m\left(\stackrel{˜}{h},\stackrel{˜}{g}\right)|{w}_{RH}\left({K}_{n}\h\right)|,$ (30)

where $\stackrel{˜}{g}$ denotes the unlabelled graph corresponding to g, $\stackrel{˜}{h}$ runs through the unlabelled subgraphs of $\stackrel{˜}{g}$ and $m\left(\stackrel{˜}{h},\stackrel{˜}{g}\right)$ is the number of ways of obtaining $\stackrel{˜}{h}$ by removing some edges in $\stackrel{˜}{h}$ . We obtain these multiplicities $m\left(\stackrel{˜}{h},\stackrel{˜}{g}\right)$ by combinatorial arguments.

4.1. The Mayer Weight of the Graph ${K}_{n}\\left(\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}\right)$

Proposition 11. For $n\ge 7$ , we have

$\begin{array}{c}|{w}_{M}\left({K}_{n}\\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}\right)|=n+\frac{14}{n-1}+\frac{48}{\left(n-1\right)\left(n-2\right)}+\frac{114}{\left(n-1\right)\cdots \left(n-3\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+\frac{156}{\left(n-1\right)\cdots \left(n-4\right)}+\frac{72}{\left(n-1\right)\cdots \left(n-5\right)}\cdot \end{array}$ (31)

Proof. The over graphs of ${K}_{n}\\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}$ whose Ree-Hoover weight is not zero are up to isomorphism of the form: ${K}_{n}\{C}_{4}$ , ${K}_{n}\{C}_{4}\cdot {S}_{l}$ , $1\le l\le 2$ , ${K}_{n}\{C}_{4}\cdot \cdot {S}_{2}$ , ${K}_{n}\\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}$ , ${K}_{n}\{S}_{l}$ , $1\le l\le 4$ , ${K}_{n}\\left({S}_{1}-{S}_{l}\right)$ , $1\le l\le 3$ , and ${K}_{n}$ . Their multiplicities are given by

$\begin{array}{l}|{w}_{M}\left({K}_{n}\\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}\right)|\\ =|{w}_{RH}\left({K}_{n}\right)|+7|{w}_{RH}\left({K}_{n}\{S}_{1}\right)|+12|{w}_{RH}\left({K}_{n}\{S}_{2}\right)|+5|{w}_{RH}\left({K}_{n}\{S}_{3}\right)|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+|{w}_{RH}\left({K}_{n}\{S}_{4}\right)|+3|{w}_{RH}\left({K}_{n}\{C}_{4}\right)|+15|{w}_{RH}\left({K}_{n}\\left({S}_{1}-{S}_{1}\right)\right)|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+12|{w}_{RH}\left({K}_{n}\\left({S}_{1}-{S}_{2}\right)\right)|+3|{w}_{RH}\left({K}_{n}\\left({S}_{1}-{S}_{3}\right)\right)|+|{w}_{RH}\left({K}_{n}\\left({C}_{4}\cdot \cdot {S}_{2}\right)\right)|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+9|{w}_{RH}\left({K}_{n}\\left({C}_{4}\cdot {S}_{1}\right)\right)|+3|{w}_{RH}\left({K}_{n}\\left({C}_{4}\cdot {S}_{2}\right)\right)|+|{w}_{RH}\left({K}_{n}\\left(\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}\right)\right)|.\end{array}$

We conclude using Propositions (5) and (6)-(10).

4.2. The Mayer Weight of the Graph ${K}_{n}\\left(\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{k}\right)$

Proposition 12. For $k\ge 1$ , $n\ge k+6$ , ${g}_{n}={K}_{n}\\left(\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{k}\right)$ , we have, with the usual convention $\left(\begin{array}{c}k+1\\ \mathcal{l}\end{array}\right)=0$ if $\mathcal{l}>k+1$ ,

$\begin{array}{c}|{w}_{M}\left({g}_{n}\right)|=n+\frac{6}{\left(n-1\right)}+\frac{60}{\left(n-1\right)\left(n-2\right)\left(n-3\right)\left(n-4\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{24}{\left(n-1\right)\left(n-2\right)}+\frac{60}{\left(n-1\right)\left(n-2\right)\left(n-3\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{l=1}{\overset{k+3}{\sum }}\left[\left(\begin{array}{c}k+3\\ l\end{array}\right)\frac{2l!}{\left(n-1\right)\cdots \left(n-l\right)}+\left(\begin{array}{c}k+1\\ l\end{array}\right)\frac{12l!}{\left(n-1\right)\cdots \left(n-l-3\right)}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{l=1}{\overset{k+2}{\sum }}\left[\left(\begin{array}{c}k\\ l-2\end{array}\right)+2\left(\begin{array}{c}k\\ l-1\end{array}\right)+\left(\begin{array}{c}k\\ l\end{array}\right)\right]\left[\frac{6l!}{\left(n-1\right)\cdots \left(n-l-2\right)}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{l=1}{\overset{k}{\sum }}\left(\begin{array}{c}k\\ l\end{array}\right)\left[\frac{12l!}{\left(n-1\right)\cdots \left(n-l-4\right)}\right]\cdot \end{array}$

Proof. The over graphs of ${K}_{n}\\left(\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{k}\right)$ whose Ree-Hoover weight is not zero are up to isomorphism of the form: ${K}_{n}\{S}_{l}$ , $1\le l\le k+3$ , ${K}_{n}\\left({C}_{4}\cdot {S}_{l}\right)$ , $1\le l\le k+1$ , ${K}_{n}\\left({S}_{1}-{S}_{l}\right)$ , $1\le l\le k+2$ , ${K}_{n}\{C}_{4}$ , ${K}_{n}\\left({C}_{4}\cdot \cdot {S}_{2}\right)$ , ${K}_{n}\\left(\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{l}\right)$ , $1\le l\le k$ and ${K}_{n}$ . Their multiplicities are given by

$\begin{array}{c}|{w}_{M}\left({g}_{n}\right)|=|{w}_{RH}\left({K}_{n}\right)|+3|{w}_{RH}\left({K}_{n}\{S}_{1}\right)|+6|{w}_{RH}\left({K}_{n}\{S}_{2}\right)|+|{w}_{RH}\left({K}_{n}\{S}_{3}\right)|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+3|{w}_{RH}\left({K}_{n}\{C}_{4}\right)|+12|{w}_{RH}\left({K}_{n}\{S}_{1}-{S}_{1}\right)|+6|{w}_{RH}\left({K}_{n}\{S}_{1}-{S}_{2}\right)|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+|{w}_{RH}\left({K}_{n}\\left({C}_{4}\cdot \cdot {S}_{2}\right)\right)|+3|{w}_{M}\left({K}_{n}\\left({C}_{4}\cdot {S}_{1}\right)\right)|\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{l=1}{\overset{k+1}{\sum }}\text{ }3\left(\begin{array}{c}k+1\\ l\end{array}\right)\left[|{w}_{RH}\left({K}_{n}\{C}_{4}\cdot {S}_{l}\right)|+\left(\begin{array}{c}k\\ l\end{array}\right)|{w}_{RH}\left({K}_{n}\\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{l}\right)|\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{l=1}{\overset{k+2}{\sum }}\text{ }3\left[\left(\begin{array}{c}k\\ l\end{array}\right)+2\left(\begin{array}{c}k\\ l-1\end{array}\right)+\left(\begin{array}{c}k\\ l-2\end{array}\right)\right]|{w}_{RH}\left({K}_{n}\{S}_{1}-{S}_{l}\right)|\end{array}$

$\text{\hspace{0.17em}}+\underset{l=1}{\overset{k+3}{\sum }}\left(\begin{array}{c}k+3\\ l\end{array}\right)|{w}_{RH}\left({K}_{n}\{S}_{l}\right)|.$

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 $\left({K}_{n}\\left(\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}\right)\right)$ , $n\ge 7$ , and more generally for the family $\left({K}_{n}\\left(\left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{k}\right)\right)$ , $k\ge 1$ , $n\ge k+6$ . 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  , for families of graphs $\left({K}_{n}\\left({C}_{4}\cdot \cdot {S}_{2}\right)\right)$ , $n\ge 7$ and $\left({K}_{n}\\left({S}_{1}\cdot {C}_{4}\cdot \cdot {S}_{2}\right)\right)$ , $n\ge 7$ , and more generally for families $\left({K}_{n}\\left({S}_{k}\cdot {C}_{4}\cdot \cdot {S}_{2}\right)\right)$ , $k\ge 1$ , $n\ge k+6$ . 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 $\left({K}_{n}\\left({S}_{1}\cdot \left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}\right)\right)$ , $n\ge 8$ and $\left({K}_{n}\\left({S}_{1}\cdot \left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{2}\right)\right)$ , $n\ge 9$ and $\left({K}_{n}\\left({S}_{2}\cdot \left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}\right)\right)$ , $n\ge 9$ and more generally for families $\left({K}_{n}\\left({S}_{1}\cdot \left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{k}\right)\right)$ , $k\ge 1$ , $n\ge k+6$ and $\left({K}_{n}\\left({S}_{k}\cdot \left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{1}\right)\right)$ , $k\ge 1$ , $n\ge k+6$ , with $\left({S}_{j}\cdot \left({C}_{4}\cdot \cdot {S}_{2}\right)\cdot {S}_{k}\right)$ denote the graph obtained by joining with an edge of the graph ${C}_{4}\cdot \cdot {S}_{2}$ the centers of a j-star and k-star.

Conflicts of Interest

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

Cite this paper

Kaouche, A. (2019) Other Formulas for the Ree-Hoover and Mayer Weights of Families of 2-Connected Graphs. Journal of Applied Mathematics and Physics, 7, 1800-1813. https://doi.org/10.4236/jamp.2019.78123

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