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%\received{15 November 2007}
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\begin{document}
%
\title[Mayer and Ree--Hoover weights of infinite families]{Mayer and Ree--Hoover weights of infinite families of $2$-connected graphs}
\author{Amel Kaouche and Pierre Leroux}
\address{LaCIM, Universit\'e du Qu\'ebec \`a Montr\'eal\\
CP 8888, Succ. Centre-ville\\
Montr\'eal (Qu\'ebec) H3C 3P8
}
%\address{\addressmark{1}LaCIM, D\'epartement de Math\'ematiques, Universit\'e du Qu\'ebec \`a Montr\'eal}
%\keywords{graph invariants, non-ideal gas, hard sphere gas, Mayer expansion, virial expansion.}

\date{}

\begin{abstract}
We study graph weights (i.e., graph invariants) which arise naturally in Mayer's theory and Ree--Hoover's theory of  virial expansions in the context of a non-ideal gas. We give special attention to the Second Mayer weight $w_M(c)$ and the Ree--Hoover weight $w_{RH}(c)$  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 $c$. Among our results are explicit formulas for the values of Mayer weights and Ree--Hoover weights for certain infinite families of $2$-connected graphs.
\vskip25pt

\noindent  {{\sc R\'esum\'e.  }}Nous \'etudions des poids de graphes (\`a savoir, des invariants de graphes) qui apparaissent naturellement dans les th\'eories de Mayer et de Ree--Hoover du d\'eveloppement du viriel dans le contexte de gas non-id\'eaux. Nous nous int\'eressons sp\'ecialement au second poids de Mayer $w_M(c)$ et de Ree--Hoover $w_{RH}(c)$ pour un graphe 2-connexe $c$ provenant d'un gaz \`a noyaux durs en dimension un. Ces poids sont calcul\'es en utilisant des volumes sign\'es de polytopes convexes associ\'es au graphe $c$. Parmi nos r\'esultats, nous donnons des formules explicites pour le poids de Mayer et de Ree--Hoover pour certaines familles infinies de graphes 2-connexes.
%\end{resume} 
\end{abstract}
\maketitle
%\pagebreak
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%First page headline in LaTeX for S\'eminaire Lotharingien de Combinatoire
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\font\rms=cmr8 
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\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 61A \rms (2009), Article~B61Af\hfill}
\def\thepage{}

% Section 0 : Introduction

\section{Introduction}

\emph{Graph weights} can be defined as functions on (simple, finite)
graphs taking scalar or polynomial values and which are invariant
under isomorphism, i.e., under vertex relabelling. Since most
graphical concepts share this invariance property, examples of graph
weights abound. For instance,   the \emph{graph complexity} $\gamma(g)$ of a graph $g$, which is defined as the number of maximal spanning forests of $g$, is an example of a graph weight.
In the context of a non-ideal gas in a vessel $V \subseteq \R^d$, the \emph{Second Mayer weight} $w_M(c)$  of a connected graph $c$, over the set $[n]=\{1, 2, \dots, n\}$ of vertices, is defined by
%
\begin{equation}
w_M(c)= \int  \nolimits_{(\R^{d})^{n-1}} \prod\limits_{\{i,j\}\in
c}f(\|\overrightarrow{x_i}-\overrightarrow{x_j}\| )  \  d\overrightarrow{x_{1}} \cdots  d\overrightarrow{x_{n-1}},\ \ \ \overrightarrow{x_n}=0,
\label{eq7}
\end{equation}
%
where $\overrightarrow{x_1}, \dots,\overrightarrow{x_n}$ are variables in $\R^d$ representing the positions of $n$ particles in  $V$
($V\rightarrow\infty$), the value $\overrightarrow{x_n}=0$ being arbitrarily fixed, and where $f=f(r)$ is real-valued function associated with the pairwise interaction potential of the particles,  see \cite{FordUhlenbeck,lld}.  

Let $\C[n]$ be the set of connected graphs over $[n].$
The total sum of weights of connected graphs over $[n]$ is denoted by
\begin{equation}
|\C[n]|_{w_M} = \sum_{c\in\C[n]} w_M(c).
\end{equation}
%
%First page headline in AmS-LaTeX for S\'eminaire Lotharingien de Combinatoire
%--restoring the headers and pagenumbering
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\addtocounter{page}{1}
\markboth{\SMALL AMEL KAOUCHE AND PIERRE LEROUX}{\SMALL 
MAYER AND REE--HOOVER WEIGHTS OF INFINITE FAMILIES}
%
%
\leavevmode
\noindent
\hskip-14pt 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 \cite{lld}):
%
\begin{equation}
\frac{P}{kT} = \C_{w_M}(z) = \sum_{n\geq1}|\C[n]|_{w_M} \frac{z^n}{n!},
\end{equation}
%
where $k$ is a constant, $T$ is the temperature and $z$ is a variable called the \emph{fugacity} or the \emph{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 $w_M(c)$ of the connected graphs $c\in \C[n]$, it is sufficient to compute the weights $w_M(b)$ for 2-connected graphs $b\in\B[n]$ ($\B$ for \emph{blocks}). Moreover,  these occur in the so-called \emph{virial expansion} proposed by Kamerlingh Onnes in $1901$
%
\begin{equation}
\frac{P}{kT}=\rho+\beta_2\rho^2+\beta_3\rho^3+\cdots ,
\end{equation}
%
where $\rho$ is the density. Indeed, it can be shown that
%
\begin{equation}
\label{ eq:bnBwM}
\beta_n=\frac{1-n}{n!}|\mathcal{B}[n]|_{w_{M}},
\end{equation}
%
where $\mathcal{B}[n]$ denote the set of $2$-connected graphs over $[n]$ and $|\mathcal{B}[n]|_{w_{M}}$ is the total sum of weights of $2$-connected graphs over $[n].$
In order to compute this expansion numerically, Ree and Hoover \cite{RH1} introduced a modified weight denoted by $w_{RH}(b)$, for 2-connected graphs $b,$ which greatly simplifies the computations. It is defined by
%
\begin{equation}
w_{RH}(b)= \int  \nolimits_{(\R^{d})^{n-1}} \prod\limits_{\{i,j\}\in
b} f(\|\overrightarrow{x_i}-\overrightarrow{x_j}\| )  \prod\limits_{\{i,j\}\notin
{b}}\overline{f}(\|\overrightarrow{x_i}-\overrightarrow{x_j}\| )    \  d\overrightarrow{x_{1}} \cdots  d\overrightarrow{x_{n-1}},\ \ \ \overrightarrow{x_n}=0,
\label{eq:w{RH}}
\end{equation}
%
where $\overline{f}(r) = 1+f(r)$.
Using this new weight, Ree and Hoover \cite{ RH1,RH2,RH3} and later Clisby and McCoy \cite{Clisby2,Clisby} have computed the virial coefficients $\beta_n$, for $n$ up to 10,  in dimensions $d\leq8$, in the case of the hard-core continuum gas, that is when the interaction is given by
 %
\begin{equation} \label{eq:fftilde}
f(r) = - \chi(r<1) ,\  \  \
\overline{f}(r)=\chi(r\geq1),
\end{equation}
%
where $\chi$ denotes the characteristic function ($\chi(P)=1,$ if $P$ is true and $0,$ otherwise).

While physicists are interested in summing the weights of all connected or 2-connected
graphs of a given order, the present paper focuses on individual
graph contributions and their combinatorial significance. For a given individual graph, the Mayer and Ree--Hoover weights can be computed using Ehrhart polynomials in the case of the hard-core continuum gas in dimension $d=1$. This has been done systematically for graphs of size $\leq 8$ in \cite{Kaouche,KLtables}. The main goal of the present paper is to give exact formulas for certain infinite families of graphs.

In Section~$1$, we use M\"obius  inversion to give explicit linear relations expressing the Ree--Hoover weights in terms of the Mayer weights and vice versa. The total Mayer weight $|\B[n]|_{w_M}$ is then rewritten in terms of the weight function $w_{RH}$ introduced by Ree and Hoover  \cite{ RH1,RH2}. This rewriting involves special coefficients called star contents. The interest of using the Ree--Hoover weight is that it has the value zero for many graphs. Some general explicit and recursive properties of the star content are given.

Section~$2$ is devoted to the special 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 $\mathcal{P}(c)$ naturally associated with the graph $c.$ Sufficient conditions for the nullity of the Ree--Hoover weight of a graph are also given.
An alternate useful tool, a decomposition of the polytope $\mathcal{P}(c)$ into a certain number of $(n-1)$-dimensional simplices, of volume $1/(n-1)!$ is exploited. This method was introduced in \cite{lld} and is called the method of graph homomorphisms. We adapt this method to the context of Ree--Hoover weights.
%
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 (e.g., the complete graphs $K_n$, the cycle graphs $C_n$). In the present paper we extend this list to graphs of the form $K_n\backslash g,$ where $g$ can be a star graph, a cycle, path graph, or connections of some of these graphs, etc. 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, $\{i,j\}\in g$ means 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. $e\in g$), $g\backslash e$ denotes the graph obtained from $g$ by removing the edge $e$. If $a$ is a vertex of $g$, $g\backslash a$ denotes the graph obtained by removing from $g$ vertex $a$ and all its incident edges. If $b$ and $d$ are graphs, $b\subseteq d$ means that $b$ is a subgraph of $d$. The complete graph on the vertex set $[n]=\{1,2,\ldots , n  \}$ is denoted by $K_n$. The complementary graph of a subgraph $g\subseteq K_n$ is the graph $\overline{g}=K_n\backslash g$.
%
%We made tables giving the values $w_{M}(c)$ and $w_{RH}(c)$ for all $2$-connected graphs $c$ of size at most 8 using Ehrhart polynomials.
%These extend a previous table of the Mayer weight for sizes at most $6.$ 
%These tables are available in electronic versions (see \cite{KLtables}).
\medskip

\noindent
{\bf Acknowledgements}. 
%We would like to acknowledge particularly the contribution of Gilbert Labelle and C\'edric Lamathe. 
The first author would like to acknowledge particularly one referee for his valuable suggestions for the improvement of the paper, and Gilbert Labelle and C\'edric Lamathe for their help.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% %%%       \section   %%%%%%%%%%%%
\section{Relations between $w_{M}$ and $w_{RH}$}
\subsection{Mayer weight, Ree--Hoover weight and star content}
%%%%%%%%%%%%%
An important rewriting of the virial coefficients was performed by Ree and Hoover \cite{RH1, RH2} by introducing the function
%
\begin{equation}
\overline{f}(r)=1+f(r)
\end{equation}
%
and defining a new weight  (denoted here by $w_{RH}(b)$) for $2$-connected graphs $b,$ by (\ref{eq1:wRH})
%
\begin{equation}
w_{RH}(b)= \int  \nolimits_{(\R^{d})^{n-1}} \prod\limits_{\{i,j\}\in
b} f(\|\overrightarrow{x_i}-\overrightarrow{x_j}\| )  \prod\limits_{\{i,j\}\notin
{b}}\overline{f}(\|\overrightarrow{x_i}-\overrightarrow{x_j}\| )    \  d\overrightarrow{x_{1}} \cdots  d\overrightarrow{x_{n-1}},\ \ \ \overrightarrow{x_n}=0,
\label{eq1:wRH}
\end{equation}
%
and then expanding each weight $w_M(b)$ by
substituting $1=\overline{f}-f$ for pairs of vertices not connected
by edges. Upon performing this rewriting of the
Mayer weights series, vertices in the resultant graphs will all be
mutually connected by either $f$ bonds (solid lines) or$\overline{f}$ bonds (dotted lines). 
%Now there is a new factor which appears for each graph, called the \emph{star content} by Ree and Hoover \cite{RH2}, which may be either positive or negative. 
For example, we have
\begin{equation}
\label{eq220}
w_{M}(\graphiqueformule{fig-K4.eps} \  \  )=  w_{RH}(\graphiqueformule{fig-K4.eps} \  \  ),
\end{equation}
\begin{equation}
\label{eq221}
w_M(\graphiqueformule{fig-K4moinsE.eps} \  \  )= w_{RH}(\graphiqueformule{fig-K4moinsE-RHBIS.eps} \  \  )-w_{RH}(\graphiqueformule{fig-K4.eps} \  \ ),
\end{equation}
\begin{align}
\label{eq222}
\notag w_M(\graphiqueformule{fig-C4.eps} \  \  )&= w_{MRH}(\graphiqueformule{fig-C4-DRH.eps} \  \  )-w_{M}(\graphiqueformule{fig-K4moinsE.eps} \  \  )\\
\notag &= \{w_{RH}(\graphiqueformule{fig-C4-RH.eps} \  \  )-w_{RH}(\graphiqueformule{fig-K4moinsE-RH.eps} \  \  )\}-\{w_{RH}(\graphiqueformule{fig-K4moinsE-RHBIS.eps} \  \  )-w_{RH}(\graphiqueformule{fig-K4.eps} \  \  ) \} \\
&= w_{RH}(\graphiqueformule{fig-C4-RH.eps} \  \  )-2\cdot w_{RH}(\graphiqueformule{fig-K4moinsE-RH.eps} \  \  )+w_{RH}(\graphiqueformule{fig-K4.eps} \  \  ).
\end{align}
%%
In the general situation, using M\"obius inversion, it is easy to state formulas linking the two weights $w_M$ and $w_{RH}.$ These formulas are implicit in the work of \cite{RH2}.
%% porp relations entre wM et wRH
\begin{proposition}
 \label{prop: relations}
For a 2-connected graph $b$, we have
%
\begin{align}
\label{eq: w_RH  et w_M} w_{RH}(b)& =  \sum_{b\subseteq d\subseteq K_n}w_{M}(d),\\
\label{eq: w_M  et w_RH} w_{M}(b) & =  \sum_{b\subseteq d\subseteq K_n}(-1)^{e(d)-e(b)}w_{RH}(d).
\end{align}
%
%%In the case of hard-core continuum gas, we also have
%%
%%\begin{equation}
%%\label{eq:2w_M et w_RH} |w_{M}(b)|=\sum_{b\subseteq d\subseteq K_n}|w_{RH}(d)|.
%%\end{equation}
%
\end{proposition}

%% preuve de la prop
\begin{proof}
From (\ref{eq1:wRH}) and the fact that $\overline{f}=1+f$, we have
\begin{align*}
\notag 
w_{RH}(b)&= \int  \nolimits_{(\R^{d})^{n-1}} \prod\limits_{\{i,j\}\in
b} f(\|\overrightarrow{x_i}-\overrightarrow{x_j}\| )  \prod\limits_{\{i,j\}\notin
{b}}\overline{f}(\|\overrightarrow{x_i}-\overrightarrow{x_j} \|)    \  d\overrightarrow{x_{1}} \cdots  d\overrightarrow{x_{n-1}}\\
\notag 
&= \int  \nolimits_{(\R^{d})^{n-1}} \prod\limits_{\{i,j\}\in
b} f(\|\overrightarrow{x_i}-\overrightarrow{x_j}\| )  \prod\limits_{\{i,j\}\in
\overline{b}}(1+{f}(\|\overrightarrow{x_i}-\overrightarrow{x_j}\| ) )   \  d\overrightarrow{x_{1}} \cdots  d\overrightarrow{x_{n-1}}\\
\notag 
&= \int  \nolimits_{(\R^{d})^{n-1}} \sum_{E \subseteq\overline{b}} \prod\limits_{\{i,j\}\in
E\cup b} f(\|\overrightarrow{x_i}-\overrightarrow{x_j}\| )    \  d\overrightarrow{x_{1}} \cdots  d\overrightarrow{x_{n-1}}\\
\notag 
&= \int  \nolimits_{(\R^{d})^{n-1}} \sum_{b \subseteq d \subseteq K_n} \prod\limits_{\{i,j\}\in
d} f(\|\overrightarrow{x_i}-\overrightarrow{x_j}\| )    \  d\overrightarrow{x_{1}} \cdots  d\overrightarrow{x_{n-1}}\\
&=\sum_{b\subseteq d\subseteq K_n}w_{M}(d).
\label{eq2:wRH}
\end{align*}
Using M\"obius inversion on (\ref{eq: w_RH  et w_M}) for the lattice of subsets of $K_n$, with the M\"obius function $\mu(b,d)=(-1)^{e(d)-e(b)},$ we find (\ref{eq: w_M  et w_RH}). 
%%Taking into account (\ref{eq:fftilde}), we can write (\ref{eq: w_M  et w_RH}) as $(-1)^{e(b)}w_{M}(b)=\sum_{b\subseteq d\subseteq K_n}(-1)^{e(d)}w_{RH}(d),$ which is  (\ref{eq:2w_M  et w_RH}). 
\end{proof}
%% fin de preuve
%% rem  2 %%
\begin{rem}
By the definition of the Ree--Hoover weight, we have in particular
%
\begin{equation} \label{eq:wM=wRH}
w_{RH}(K_n)=w_{M}(K_n), \quad n\geq 2.
\end{equation}
%
\end{rem}
%
When expressing the total weight $|\B[n]|_{w_M}$ in terms of Ree--Hoover weights, there is a new factor which appears for each graph in the expansion of each Mayer weight $w_M(b)$. This factor is called the \emph{star content} by Ree and Hoover \cite{RH2}, and may be either a positive or negative integer. 
%
%% def star
\begin{definition}

Let $d$ be a $2$-connected graph over the set $[n]=\{1,2,\cdots,n\}.$ Then, ${a}_n(d),$ the star content of the graph $d$, is defined by
\begin{equation}
\label{eq:a_n }
{a}_n(d)=\underset{b \in \mathcal{B}[n]}{\underset{b\subseteq d}{\sum}}(-1)^{e(d)-e(b)}.
\end{equation}
\end{definition}

%% fin def
%% prop B[n]
\begin{proposition}

$|\mathcal{B}[n]|_{w_{M}},$ the total sum of weights of $2$-connected graphs over $[n]$ is given by
%
\begin{equation}
 |\mathcal{B}[n]|_{w_{M} }=\sum_{d \in \mathcal{B}[n]}{a}_n(d)w_{RH}(d). \label{eq:B[n]}
\end{equation}
%
\end{proposition}

%% fin prop
%% preuve
\begin{proof}
Using (\ref{eq: w_M  et w_RH}) we can write $|\mathcal{B}[n]|_{w_{M}}$ as
%
\begin{align*}
\notag |\mathcal{B}[n]|_{w_{M}}&=\sum_{b \in \mathcal{B}[n]}w_{M}(b)\\
\notag&= \sum_{b \in \mathcal{B}[n]}\sum_{b\subseteq d\subseteq K_n}(-1)^{e(d)-e(b)}w_{RH}(d)\\
 &= \sum_{d \in \mathcal{B}[n]}\underset{b \in \mathcal{B}[n]}{\underset{b\subseteq d}{\sum}}(-1)^{e(d)-e(b)}w_{RH}(d)\\
 &= \sum_{d \in \mathcal{B}[n]}{a}_n(d)w_{RH}(d).\label{eq2:a_n(b)}
\end{align*}
%
\end{proof}
%% fin preuve
So, we can write $\beta_{n}$ as
%
\begin{equation}
 \beta_n=\frac{1-n}{n!}\sum_{b \in \mathcal{B}[n]}{a}_n(b)w_{RH}(b). \label{eq:B_n}
\end{equation}
%
Note that since $w_M$ and $w_{RH}$ are graph invariants, the sum (\ref{eq:B[n]}) can be simplified as
%
\begin{align}
 |\mathcal{B}[n]|_{w_{M} }
&=\nonumber
\sum_{\widetilde{b}\in\widetilde{\B}[n]}\ell(b)w_M(b)\\
&=\label{eq20}
 \sum_{\widetilde{b} \in \widetilde{\B}[n]}\ell(b){a}_n(b)w_{RH}(b),
\end{align}
%
where $\widetilde{\B}[n]$ is the set of unlabelled graphs $\widetilde{b}$ with $n$ vertices and $\ell(b)$ is the number of labellings of $\widetilde{b}$. 

%%%%%%%%%%%  suite de l exemple
For example using the three equations (\ref{eq220})--(\ref{eq222}) and (\ref{eq20}), the total weight $|\mathcal{B}[4]|_{w_{M} }$ may be written as
\begin{align}
\label{eq22}
\notag  |\mathcal{B}[4]|_{w_{M} } &= 1\cdot w_{M}(\graphiqueformule{fig-K4.eps} \  \  )
+ 6 \cdot w_M(\graphiqueformule{fig-K4moinsE.eps} \  \  )
+ 3 \cdot w_M(\graphiqueformule{fig-C4.eps} \  \  )\\
  \notag&=   ( -2 ) \cdot w_{RH}(\graphiqueformule{fig-K4.eps} \  \  )+0   \cdot w_{RH}(\graphiqueformule{fig-K4moinsE-RH.eps} \  \  )
   + 3 \cdot w_{RH}(\graphiqueformule{fig-C4-RH.eps} \  \  )\\
 &= 1 \cdot( -2 ) \cdot w_{RH}(\graphiqueformule{fig-K4.eps} \  \  )+ 6 \cdot 0   \cdot w_{RH}(\graphiqueformule{fig-K4moinsE-RH.eps} \  \  ) + 3 \cdot 1\cdot w_{RH}(\graphiqueformule{fig-C4-RH.eps} \  \  ),
    \end{align}
where 1, 6, 3 represent $\ell(b)$ and (-2), 0, 1 are the star contents $a_4(b)$ of the graphs $b=$\graphiqueformule{fig-K4.eps}\ , \graphiqueformule{fig-K4moinsE-RH.eps}\ , \graphiqueformule{fig-C4-RH.eps}\ , respectively. 
%% prop
\begin{proposition}
 We have
\begin{equation}
\label{eq:a_n=1}
 \sum_{d \in\mathcal{B}[n]}{a}_n(d)=1,
\end{equation}
where $\mathcal{B}[n]$ is the set of all $2$-connected graphs over $[n].$
\end{proposition}

%
%% preuve
\begin{proof}
We have
\begin{align*}
\notag 
 \sum_{d \in\mathcal{B}[n]}{a}_n(d)&= \sum_{d \in\mathcal{B}[n]}\underset{b \in\mathcal{B}[n]}{\underset{b\subseteq d}{\sum}}(-1)^{e(d)-e(b)}\\
&=\sum_{b \in\mathcal{B}[n]}\sum_{b\subseteq d\subseteq K_n}(-1)^{e(d)-e(b)}.
%\\
%&=\sum_{K_n\subseteq d\subseteq K_n}(-1)^{e(d)-e(K_n)}+\underset{b\neq K_n}{\underset{b\in\mathcal{B}[n]}{\sum}}\sum_{b\subseteq d\subseteq K_n}(-1)^{e(d)-e(b)}\\
%&=1+\underset{b\neq K_n}{\underset{b\in\mathcal{B}[n]}{\sum}}\sum_{\emptyset \subseteq c\subseteq K_n\backslash b}(-1)^{e(c)}\\
%&=1.
\end{align*}
Now, for any two graphs $b,d\in\B[n]$, $b\subseteq d\subseteq K_n$ if and only if $d\backslash b\subseteq K_n\backslash b$. Hence, by the binomial theorem 
\begin{align*}
\sum_{b\subseteq d\subseteq K_n}(-1)^{e(d)-e(b)}
&= 
\sum_{g\subseteq K_n\backslash b}(-1)^{e(g)}\\
&= 
(1+(-1))^{e(K_n \backslash b)}
\end{align*}
is equal to 1 if $b=K_n$ and 0 otherwise. This concludes the proof of (\ref{eq:a_n=1}). 
\end{proof}
\vspace{-.35cm}
%% fin preuve
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                 Star content of some special graphs                         %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Star content of some special graphs}
This section is devoted to the study of the star content of some special graphs.
We say that a simple graph $g$ over $[n]$ is an \emph{extension} of another graph $h$ over $[n-1],$ if $g$ is obtained from $h$ by adding a new vertex $n$ along with some edges between $n$ and the graph $h.$ Let $g \big|_{[n-1]}= g\backslash n$ denote the restriction of $g$ to the set $[n-1]$ (that is, the vertex $n$ and all its incident edges are removed).
%% prop
\begin{proposition}
\label{recurrence}
Let $g$ a $2$-connected graph over $[n]$ such that the vertex $n$ is joined to every other vertex and let $g'=g \backslash n$ be the graph obtained from $g$ by removing the vertex $n$ and all its incident edges. Then if $g'$ is also $2$-connected, we have
%
\begin{equation}
\label{eq:a_n+1}
{a}_{n}(g)=(-1)^{n-1}(n-2) \ {a}_{n-1}(g').
\end{equation}
%
\end{proposition}

In order to prove this proposition, we need first the three following lemmas:
%% lem 1
\begin{lem}\label{cutpoint}
Let $e=\{a,b\}$ be an edge of a $2$-connected graph $G$ where the vertex $a$ is a cutpoint of the graph $G\backslash b.$ Then, $G\backslash e$ is $2$-connected. 
\end{lem}
%\begin{figure}[ht]
%\begin{center}
%\scalebox{.55}{\input{ComposantesConnexes.pstex_t}}
%\caption{A $2$-connected graph $G$ with the connected components of $G\backslash b\backslash a$}
%\label{fig:S3}
%\end{center}
%\end{figure} 
\noindent
%
% proof lem 1
\begin{proof}
%It is sufficient to show that the graph $G\backslash e$ does not have any cutpoint.
%Since $G$ is $2$-connected, then $G\backslash b=G\backslash e\backslash b$ and $G\backslash a=G\backslash e\backslash a$ are connected, so $a$ and $b$ are not  cutpoints of $G\backslash e.$
%For any other vertex $c,$ $G\backslash c$ is always connected and $G\backslash e\backslash c$ is also connected. Indeed, since $a$ is a cutpoint of $G\backslash b$ and $G$ is $2$ connected, then $a$ and $b$ are connected in $G$ with the connected components of $G\backslash b\backslash a$ (see Figure~\ref{fig: ComposantesCon}). Thus $G\backslash e$ is $2$-connected. 
Since $G$ is 2-connected, both $a$ and $b$ are connected to each connected components $C_1,\ldots ,C_k$ of $G\backslash a\backslash b$ (see Figure~\ref{fig: ComposantesCon}). Moreover, since the vertex $a$ is a cutpoint , the number $k$ of components is at least 2. Consider now a vertex $c$ of $G$. We have to prove $G\backslash c\backslash e$ is connected. Since $G$ is 2-connected, we know that $G\backslash c$ is connected. Now if $c$ is equal to $a$ or $b$, the graph $G\backslash c\backslash e=G\backslash c$ is connected. If $c\neq a,b$ the vertex $c$ belongs to one of the components $C_1,\ldots ,C_k$, say $C_1$. Moreover $a$ et $b$ are linked by a path of $G\backslash c$ going through $C_2$. Hence $e=\{a,b\}$ is not an isthmus of $G\backslash c$ and $G\backslash c\backslash e$ is connected.  
\end{proof}
%
 %% figure 
\begin{figure}[H]
\vspace{.2cm}
\centerline{\includegraphics[width=.35\textwidth]{ComposantesConnexes.eps}}
 \caption{A $2$-connected graph $G$ with the connected components of $G\backslash b\backslash a$}
 \label{fig: ComposantesCon}
 \end{figure}
 %
%% lem 2
\begin{lem}\label{Inv_1}
Let $h$ be a $2$-connected graph over $[n-1].$ Then we have
\begin{equation}
\label{eq: e(g)-e(h)}
\underset{g\backslash n= h}{\underset{g \in\mathcal{B}[n]}{\sum}}(-1)^{e(g)-e(h)}=n-2\cdot
\end{equation} 
\end{lem}
% proof lem 2
\begin{proof}
For a $2$-connected graph $h$ over the set $[n-1],$ consider the set
\begin{equation*}
\label{eq:A}
A=\{g \ | \ g \  \mathrm{is} \  \mathrm{a} \ \mathrm{graph} \ \mathrm{over} \ [n] \ \mathrm{and} \  g\backslash n=h \},
\end{equation*} 
and define a weight function $w$ over $A$ by
\begin{equation*}
\label{eq: w(g)}
w(g)=(-1)^{e(g)-e(h)}, \quad  \forall g \in A.
\end{equation*}
Consider the following involution $ \mathrm{Inv_1}$ acting on the set  $A\!:$
\begin{equation*}
  \mathrm{Inv_1}(g)=\begin{cases}
 g\backslash \{1,n\}, & \text{if }   \{1,n\} \in g, \\
g\cup \{1,n\}, &  \text{otherwise}. 
\end{cases}
\label{eq:Inv_1}
\end{equation*}
The total weight of the set $A,$ $|A|_w,$ is given by
\begin{equation*}
|A|_w= \sum_{g \in A}(-1)^{e(g)-e(h)}=|\mathrm{Fix \ Inv_1}|_w,
\end{equation*}
 where $\mathrm{Fix \ Inv_1}$ denotes the set of all fixed points of the involution $\mathrm{Inv_1}.$ 
It is easy to see that $\mathrm{Inv_1}$ has no fixed point. Hence,
\begin{equation}\label{eq:|A|_w}
|A|_w= \sum_{g\backslash n= h}(-1)^{e(g)-e(h)}=0.
\end{equation} 
Moreover, there are $n-1$ different ways to obtain an extension $g$ by adding a unique edge between $n$ and $h$ for any $2$-connected graph $h$ of size $n-1.$ In this case, $g$ is not $2$-connected. In the same way, there is only one extension $g$ for $h$ by adding the vertex $n$ without new edge and therefore $g$ is also not connected. If we add the vertex $n$ and two or more edges to the graph $h,$ $g$ is always $2$-connected.
Then we can write (\ref {eq:|A|_w}) as 
\begin{eqnarray*}
\notag \underset{g\backslash n= h}{\underset{g \in\mathcal{B}[n]}{\sum}}(-1)^{e(g)-e(h)} & = &- \underset{g\backslash n= h}{\underset{g \notin\mathcal{B}[n]}{\sum}}(-1)^{e(g)-e(h)} \\
\notag & = & -(-(n-1)+1)\\
  & = & n-2,
 \end{eqnarray*}
 which concludes the proof.
 \end{proof}
 % lem 3
 \begin{lem}\label{Inv_2}
Let $h$ be a graph over the set $[n-1]$ which is not $2$-connected. Then we have
\begin{equation}
\label{eq: e(g)}
\underset{g\backslash n= h}{\underset{g \in\mathcal{B}[n]}{\sum}}(-1)^{e(g)}=0\cdot
\end{equation} 
\end{lem}
% proof lem 3
\begin{proof}
For a graph $h$ over the set $[n-1]$ which is not $2$-connected, consider the set
\begin{equation*}
\label{eq:B}
B=\{g \ | \ g \  \mathrm{is} \  \mathrm{a} \ 2\mathrm{-connected} \  \mathrm{graph}  \ \mathrm{over} \ [n] \ \mathrm{and} \  g\backslash n=h \}.
\end{equation*} 
If $h$ is not connected then $B=\emptyset$ and (\ref{eq: e(g)}) is trivially satisfied. If $h$ is connected let $a$ be the cutpoint of $h$ having the smallest label.
Define the following weight function $w$ over $B:$
\begin{equation*}
\label{eq: w-B(g)}
w(g)=(-1)^{e(g)}, \quad  \forall g \in B.
\end{equation*}
Consider the following involution $\mathrm{Inv_2}$ acting on the set  $B:$ 
\begin{equation*}
\mathrm{Inv_2}(g)=\begin{cases}
 g\backslash \{a,n\}, & \text{if }  \{a,n\} \in g, \\
g\cup \{a,n\}, &  \text{otherwise}. 
\end{cases}
\label{eq:Inv_2}
\end{equation*}
Note that for all $g\in B$, $\mathrm{Inv_2}(g)$ is $2$-connected by Lemma~\ref{cutpoint}. Hence, $B$ is stable by the involution $\mathrm{Inv_2}$.
We deduce that
\begin{equation*}
|B|_w= \sum_{g \in B}(-1)^{e(g)}=|\mathrm{Fix \ Inv_2}|_w,
\end{equation*} 
 where $\mathrm{Fix \ Inv_2}$ denotes the set of all fixed points of the involution $\mathrm{Inv_2}.$ 
Again, the involution $\mathrm{Inv_2}$ has no fixed point. Hence,\begin{equation*}\label{eq:|B|_w}
|B|_w= \sum_{g\backslash n= h}(-1)^{e(g)}=0\cdot
\end{equation*} 
\end{proof}
 %
 We are now able to prove Proposition~\ref{recurrence}.
 % proof prop
\begin{proof}[Proof of Proposition~\ref{recurrence}]
Let $g$ a be $2$-connected graph over $[n]$ such that the vertex $n$ is joined to every other vertex and let $g'=g \backslash n$. From (\ref{eq:a_n }) we have successively
 %
\begin{align*}
{a}_{n}(g)&=\underset{h \in \mathcal{B}[n]}{\underset{h\subseteq g}{\sum}}(-1)^{e(h)-e(g)}\\
%
&=\underset{h \in \mathcal{B}[n]; h\backslash n\in \mathcal{B}[n-1]}{\underset{h\subseteq g}{\sum}}(-1)^{e(h)-e(g)}+\underset{h \in \mathcal{B}[n]; h\backslash n\notin \mathcal{B}[n-1]}{\underset{h\subseteq g}{\sum}}(-1)^{e(h)-e(g)}\\
%
&=\underset{h' \in \mathcal{B}[n-1]}{\underset{h' \subseteq g'}{\sum}} (-1)^{e(g)-e(h')} \quad \ \underset{h\backslash n=h'}{\underset{h\in \mathcal{B}[n]}{\sum}}(-1)^{e(h)-e(h')}\\
%
& \quad  \quad  \quad  \quad \quad  \quad \quad \quad \quad \quad \quad  \  \ +\underset{h' \notin \mathcal{B}[n-1]}{\underset{h'\subseteq g'}{\sum}} (-1)^{e(g)} \quad \underset{h\backslash n=h'}{\underset{h\in \mathcal{B}[n]}{\sum}}(-1)^{e(h)}\\
%
&=(-1)^{n-1}\underset{h \in \mathcal{B}[n-1]}{\underset{h'\subseteq g'}{\sum}}(-1)^{e(g')-e(h')} (n-2)+0\quad (\mbox{by} \ \mbox{Lemmas} \  \ref{Inv_1} \ \ \mbox{and}  \ \ref{Inv_2})\\
&=(-1)^{n-1}(n-2){a}_{n-1}(g') \cdot
 \end{align*}
%
 \end{proof}
% cor
\begin{cor}
Let $n> m$ and let $g$ be a $2$-connected graph over $[n]$ such that for $i=0, \ldots, n-m-1,$ vertex $n-i$ is joined to every vertex $j< n-i,$ and let
$ g^{(n-m)}=g\backslash n\backslash n-1 \backslash \cdots \backslash m+1$
be the graph obtained from $g$ by removing the vertices $n, n-1, \ldots, m+1$ and all their incident edges. Then if $g^{(n-m)}$ is also $2$-connected,
%
\begin{equation}
\label{eq:a_m}
{a}_n(g)=(-1)^{\binom {n} {2}-\binom {m}
{2}}\frac{(n-2)!}{(m-2)!}\ {a}_{m}(g^{(n-m)}).
\end{equation}
%
\end{cor}
% rem
\begin{rem}
In particular, taking $g=K_n$ and $m=2$ in \em{(\ref{eq:a_m})} we have, for $n> 2,$ 
\begin{equation}
\label{eq27 }
   {a}_{n}(K_n)=(-1)^{\binom {n} {2}+1}(n-2)!
 \end{equation}
\end{rem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       The hard-core continuum gas in one dimension              %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Hard-core continuum gas in one dimension}
Consider $n$ hard particles of diameter 1 on a line segment.  The \emph{hard-core} constraint translates into the interaction potential $\varphi$, 
%$\chi(|x_i-x_j|\geq 1)$ (
with $\varphi(r)= \infty$, if $r< 1$, and $\varphi(r)= 0$, if $r\geq 1$, and the Mayer function $f$ and the Ree--Hoover function
$\overline{f}$ are given by (\ref{eq:fftilde}).
%
%\begin{equation}
%\label{10 }
%f_{ij} =  -\chi(|x_{i}-x_{j}|< 1), \ \ \  \overline{f}_{ij}=1+f_{ij}=\chi(|x_{i}-x_{j}|\geq1).
%\end{equation}
%
Hence, we can write the Mayer weight function $w_M(c)$ of
a connected graph $c$ as
%
\begin{equation} \label{eq:wm}
w_M(c)=(-1)^{e(c)}\int \nolimits_{\R^{n-1}}
\prod\limits_{\{i,j\}\in c }\chi(|x_{i}-x_{j}|<1)d
{x_{1}}\ldots d{x_{n-1}},\ \ \ \ x_n=0,
\end{equation}
and the Ree--Hoover weight function $w_{RH}(c)$ of a 2-connected
graph $c$ as
%
\begin{equation}
\label{eq:wRH} w_{RH}(c)=(-1)^{e(c)}\int \nolimits_{\R^{n-1}}
\prod\limits_{\{i,j\}\in c
}\chi(|x_{i}-x_{j}|<1)\prod\limits_{\{i,j\}\notin
{c}}\chi(|x_{i}-x_{j}|>1)d {x_{1}}\ldots d{x_{n-1}},
\end{equation}
%
with $x_n=0$ and where $e(c)$ is the number of edges of $c$.
%%  de 1+2 lemm 1 pour wRH =0
Note that $w_M(c)= (-1)^{e(c)} \Vol(\mathcal{P}(c)),$ where $\mathcal{P}(c)$ is the polytope defined by 
%
\begin{equation*}
%\label{eq17} 
\mathcal{P}(c)=\{ X \in \mathbb{R}^{n}\ | \ x_ n =0
, |x_{i}-x_{j}|< 1 \hspace*{0.1 cm}\forall
\{i,j\}\in c \} \subseteq \R^{n-1} \times \{0\}\subseteq \R^n,
\end{equation*}
%
where $X=(x_1, \ldots, x_n).$ Similarly, $w_{RH}(c)= (-1)^{e(c)} \Vol(\mathcal{P}_{RH}(c)),$ where $\mathcal{P}_{RH}(c)$ is the union of polytopes defined by
%
\begin{eqnarray*}
%\label{eq17} 
\mathcal{P}_{RH}(c)=\{ X \in \mathbb{R}^{n}\ | \ x_ n =0
,&&|x_{i}-x_{j}|< 1 \hspace*{0.1 cm}\forall
\{i,j\}\in c, \\
%
&& |x_{i}-x_{j}|> 1 \hspace*{0.1 cm}\forall \{i,j\}\in \overline{c}\} \subseteq \R^{n-1} \times \{0\}\subseteq \R^n.
\end{eqnarray*}
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   Why many graphs have Ree--Hoover weight equal to $0$ %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Why many graphs have Ree--Hoover weight equal to $0$}
When the Ree--Hoover transformation is made, many graphs have
zero star content and hence do not contribute to the virial
coefficient. In addition, some Ree--Hoover graph weights may be zero
for geometrical reasons. We found sufficient conditions for families of graphs which guaranty the nullity of their Ree--Hoover weights. We introduce first some variants of the notion of subgraph and an associated lemma:
% def
\begin{definition}
\label{def:tildagraph}
Let $g$ be a simple graph on the vertex set $U$ and  $g'$ be a subgraph of $g$ on the vertex set $U'\subseteq U.$ The graph $g'$ is said to be  {\em induced} by $g$ if 
%
 %
  \begin{equation}
\label{ }
g'=g\cap K_{U'},
\end{equation}
  %
where $K_{U'}$ is the complete graph on $U'.$ If a graph $h$ is isomorphic to an induced subgraph of $g,$ we write $h \ \overline{\subseteq} \ g.$
%\begin{itemize}
%  \item[1.]  We say that $h$ is a \em{tilda subgraph} of $g,$ and we write $h \widetilde{\subseteq} g,$ if $h$ is isomorphic to a subgraph of $g,$ that is, there exists a graph monomorphism
%  %
%  \begin{equation}
%\label{ }
%j : h \hookrightarrow g.
%\end{equation}
%  %
% \item[2.] Moreover, we say that $h$ is \em{induced} by $g$, and we write  $h\overline{\subseteq} g,$ if $h \widetilde{\subseteq} g$ that the monomorphism $j$ is such that 
% %
%  \begin{equation}
%\label{ }
%(K_{V'}\backslash h')\cap g=\emptyset,
%\end{equation}
%  %
%where $h'=j(h)$ and $V'$ is the vertex set of $h'.$
%\end{itemize}
%
\end{definition}

%%%%%%%%%%%%%%%%%
% lem
%\begin{lem}\label{lem:subgraph} 
%Let $g$ {\rm(}resp. $h${\rm)} be a simple graph on the vertex set $U$ {\rm(}resp. $V${\rm)}. Then we have
%%
%\begin{equation}\label{eq:sugraph}
%h \ \overline{\subseteq} \ g \Leftrightarrow K_{V}\backslash h \ \overline{\subseteq} \ K_{U}\backslash g.
%\end{equation}
%
%\end{lem}
%% preuve du lem
%\begin{proof}
%This follows directly from the definition.
%\end{proof}
%%

%%%%%%%%%%%%%%%%%%
% lem
%\begin{lem}\label{lem:tildagraph}
%Let $C_n,$ $n\geq5,$ be the (unoriented) $n$-cycle. Then
%\begin{equation}\label{eq:tildagraph}
%C_n\widetilde{\subseteq}  K_{n}\backslash C_n.
%\end{equation}
%\end{lem}
%% preuve lem
%\begin{proof}
%Let $ K_{n}\backslash C_n= K_{n}\backslash \{\{1,2\},\{2,3\},\ldots,\{n-1,n\},\{n,1\}\}.$
%We distinguish two cases:\\
%$\bullet$ Case 1: if $n=2k+1$ (see Figure~\ref{fig:C7} with $n=7$), we can take the cycle
%$$\{\{1,3\},\{3,5\},\ldots,\{2k-1,2k+1\},\{2k+1,2\},\{2,4\},\ldots\{2k,1\}\}.$$ 
%%%% figure 
%\begin{figure}[H]
%\begin{center}
%\scalebox{.55}{\input{fig-C7.pstex_t}}
%\caption{$C_7\widetilde{\subseteq}  K_{7}\backslash C_7$}
%\label{fig:C7}
%\end{center}
%\end{figure} 
%% fin fig
%$\bullet$ Case 2: if $n=2k$ (see Figure~\ref{fig:C8} with $n=8$), we can take the cycle
%$$\{\{1,3\},\{3,5\},\{5,7\},\{7,2\},\{2,2k\},\{2k,2(k-1)\},\ldots,\{6,4\},\{4,1\}\}.$$
%%% figure
%\begin{figure}[H]
%\begin{center}
%\scalebox{.55}{\input{fig-C8.pstex_t}}
%\caption{$C_8\widetilde{\subseteq}  K_{8}\backslash C_8$}
%\label{fig:C8}
%\end{center}
%\end{figure}
%%% fin fig
%\end{proof}
%%fin preuve
%% prop
\begin{proposition}
\label{pro:wRH(h)=0}
Let $g$ and $h$ be two $2$-connected graphs. In the case of hard-core continuum gas in one dimension, we have:
%
\begin{equation}\label{eq:wRH(h)=0}
h\ \overline{\subseteq} \ g \quad {\rm and} \quad w_{RH}(h)=0 \quad {\rm implies} \quad w_{RH}(g)=0.
\end{equation}
%
\end{proposition}

%% preuve
\begin{proof}
Without loss of generality, we can suppose that $h$ is a subgraph of $g$ where the vertex set of $h$ is $[m]$ and the one of $g$ is $[n].$ By hypothesis, $w_{RH}(h)=0,$ that is, the system $S_h$ of inequalities
%
\begin{eqnarray*}
                &|x_i-x_j|<1 & \quad {\rm for} \quad \{i, j \} \in h \\
{\rm and} & |x_i-x_j|>1 & \quad {\rm for} \quad \{i, j \} \in K_m \backslash h
\end{eqnarray*}
%
is contradictory. Consider now the system $S_g,$
%
\begin{eqnarray*}
                &|x_i-x_j|<1 & \quad {\rm for} \quad \{i, j \} \in g \\
{\rm and} & |x_i-x_j|>1 & \quad {\rm for} \quad \{i, j \} \in K_m \backslash g.
\end{eqnarray*}
%
The system $S_g$ has more ``$<$'' constraints and more ``$>$'' constraints than $S_h$. Thus, $S_g$ must be also contradictory. That is  $w_{RH}(g)=0.$
\end{proof}
%%fin preuve
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%       The hard-core continuum gas in one dimension              %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Sufficient conditions for $w_{RH}=0$}
%%%  theo wRH =0
\begin{theorem}
\label{theo:wRH=0}
The Ree--Hoover weight of a $2$-connected graph $g$ of size $n$ is zero if $g$ satisfies one of the following conditions,
\begin{equation}
\label{eq1:wRH=0}
\quad \quad \quad \quad \ g\ {\rm is\ not\ chordal}\ :\ \exists \ k \geq 4, \ \ C_k \ \overline{\subseteq} \  g, 
\end{equation}
%\item 
\begin{equation}
\label{eq2:wRH=0}
%{\rm or}\ g\ {\rm is\ claw\mathrm{\tiny-}free}\ :\ S_3 \ \overline{\subseteq} \  g,
{\rm or}\ g\ {\rm has\ a\ claw}\ :\ S_3 \ \overline{\subseteq} \  g,
\end{equation} 
where $S_3$ is the 3-star graph (see Figure~\ref{fig:S3}).
\end{theorem}

%% fin theor
%% preuve 
\begin{proof}
\begin{itemize}
\item In order to prove (\ref{eq1:wRH=0}) it is sufficient, by Proposition~\ref{pro:wRH(h)=0}, to show that $w_{RH}(C_k)=0,$ for $k \geq 4,$ where $C_k$ is the standard cycle graph $\{ \{ 1, 2\},\{ 2, 3\},\\ \ldots, \{ k-1, k\},\{ 1, k\}\}.$ Therefore, the associated system of inequalities is contradictory. Without loss of generality, we can assume that $x_1 \leq x_2$. Then $x_2 < x_3$ since $|x_1 - x_2|\leq 1,$ $|x_2 - x_3|\leq 1$ and $|x_1 - x_3|\geq 1.$ For the same reason $x_i < x_{i+1}$ for all $i=1, \ldots, k,$ with the convention that $x_{k+1}=x_1.$ So we obtain $x_1 \leq x_2 <x_3< \ldots < x_k <x_1$ which is a contradiction.

\item In order to prove (\ref{eq2:wRH=0}) it is sufficient, by Proposition~\ref{pro:wRH(h)=0}, to show that $w_{RH}(S_3)=0$ where $S_3=\{ \{ 1, 4\},\{ 2, 4\},\{ 3, 4\}\}$. Without loss of generality, we can assume that $x_4=0,$ $x_1 \leq x_2 \leq x_3.$ Since $|x_i|=|x_i - x_4|< 1,$ for $i=1,3,$ we have $|x_3 - x_1|< 2.$ This is incompatible with the conditions $|x_1 - x_2|> 1,$ $|x_2 - x_3|> 1,$ since $|x_3-x_1|=(x_3 - x_2)+(x_2 - x_1)>2.$
\end{itemize}
\end{proof}
%% fin proof
%For the proof of this theorem, see Section~\ref{Proof of theorem1}.
%Using Ehrhart polynomials, we have computed the weights $w_M(b)$ and $w_{RH}(b)$ of all 2-connected graphs of sizes 7 and 8 and we made a program (see \cite{Kaouche}) for computing in a few minutes the weights $w_{RH}(b)$ of every  2-connected graphs of size up to 13. An electronic version of the tables for sizes up to 8 are available from the authors (see \cite{KLtables}). 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% %%%       \subsection   %%%%%%%%%%%%
%%%%%%%%%%%%%%%%%% \subsection{Graph homomorphisms} %%%%%%%%%%%%%%
\subsection{The method of graph homomorphisms adapted to the Ree--Hoover weight} \label{section:graphhomo}
%In order to evaluate the volume of the polytope $\Po(b)$ associated to a $2$-connected graph $b,$ we can decompose it into $\nu(b)$ subpolytopes which are all simplices of volume $1/(n-1)!$. As noticed by Bodo Lass \cite {Lass} (see also Bernardi \cite{Bernardi}), we can obtain these subpolytopes by fixing the integral parts and the relative positions of the fractional parts of the coordinates $x_1,\dots,x_{n}$ of points $X\in\Po(b)$. The number of such configurations then yield $\nu(b)$ and we have $\Vol(\Pob) = \nu(b)/(n-1)!$.

The method of graph homomorphisms was introduced by Labelle, Leroux and Ducharme \cite{lld} for the exact computation of the Mayer weight $w_M(b)$ of an arbitrary 2-connected graph $b$ in the context of hard-core continuum gases in one dimension. Since $w_M(b)=(-1)^{e(b)}{\rm Vol}(\Po(b))$, the computation of $w_M(b)$ is reduced to the computation of the volume of the polytope $\Po(b)$ associated to $b$. In order to evaluate this volume, the polytope $\Po(b)$ is decomposed into $\nu(b)$ simplices which are all of volume $1/(n-1)!$. This yields $\Vol(\Pob) = \nu(b)/(n-1)!$. The simplices are encoded by a diagram associated to the integral parts and the relative positions of the fractional parts of the coordinates $x_1,\ldots ,x_n$ of points $X\in\Po(b)$.

More precisely, to each real number $x,$ they associate an ordered pair  $(\xi_x,h_x)$, called the fractional representation of $x$, where  $h_x=\lfloor x\rfloor$ 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 $x\neq y$, the condition $|x-y|< 1$ translates into ``assuming $\xi_x<\xi_y$, then $h_x=h_y$ or $h_x=h_y+1$". Geometrically, the slope of the line segment between the points $(\xi_x,h_x)$ and $(\xi_y,h_y)$ in the plane should be either null or negative. 
Now consider a $2$-connected graph $b$ with vertex set $V=[n]=\{1,2,\dots,n\}$, and let $X=(x_1,\dots,x_{n})$ be a point in the polytope $\Pob$.  Let us write $(\xi_i,h_i)$ for the fractional representation of the coordinate $x_i$ of $X$, $i=1,\dots,n$. For $x_n=0,$ it will be convenient to use the special representation $\xi_n=1.0$ and $h_n=-1.$
The volume of $\Pob$ is not changed by removing all hyperplanes $\{x_i-x_j=k\}$, for $k\in \Z$. Hence, we can assume that all the fractional parts $\xi_i$ are distinct. We form a subpolytope of $\Pob$ by keeping the ``heights" $h_1,h_2,\dots,h_{n}$ fixed as well as the relative positions (total order) of the fractional parts $\xi_1,\xi_2,\dots,\xi_{n}$. Let $h:V\rightarrow\Z$ denote the height function $i\mapsto h_i$ and  $\beta:V\rightarrow[n]$ be the permutation of $[n]$ for which $\beta(i)$ gives the rank of $\xi_i$ in this total order. Note that $\beta(n)=n$. The corresponding simplex will be denoted by $\Po(h,\beta)$. Explicitly, each simplex can be written as
\begin{equation}\label{eq:diese}
\Po(h,\beta) = \{ (h_1+\xi_1,\ldots ,h_{n-1}+\xi_{n-1},0)\ |\ 0<\xi_{\beta^{-1}(1)}<\cdots <\xi_{\beta^{-1}(n-1)}<1\}
\end{equation}
and it is shown in \cite{lld} (see also \cite{Kaouche} for more details) that each such simplex is affine-equivalent (with jacobian 1) to the standard simplex
\[
\Po(0,{\rm id}) = \{ (\xi_1,\xi_2,\ldots ,\xi_{n-1},0)\ |\ 0<\xi_{1}<\xi_2<\cdots <\xi_{n-1}<1\}
\]
in $\R^{n-1}\times \{0\}$, of volume $1/(n-1)!$.

Note that the simplices (\ref{eq:diese}) are disjoint and each such simplex can be characterized by its centre of gravity
\[
X_{h,\beta}=(h_1+\frac{\beta(1)}{n},h_2+\frac{\beta(2)}{n},\ldots , h_{n-1}+\frac{\beta(n-1)}{n},0)\in\R^{n-1}\times \{0\}.
\]
Note also that when there are no restrictions on $h$ and $\beta$, the union of the closed simplices $\overline{\Po(h,\beta)}$ coincides with the whole configurations space $\R^{n-1}\times \{0\}$.

Using the fractional coordinates to represent the center of gravity $X_{h,\beta}$ of the simplex $\Po(h,\beta)$, and drawing a line segment form $x_i=(h_i,\xi_i)$ and $x_j=(h_j,\xi_j)$ for each edge $\{i,j\}$ of the graph $b$, we obtain a configuration in the plane which can be seen as an homomorphic image of $b$ and which characterizes the subpolytope $\Po(h,\beta)$. For example, take $n=6$ and $$b=\{ \{1, 3\},\{1, 5\},\{1, 6\},\{2, 3\},\{2, 4\},\{5, 6\} \}.$$ Figure~\ref{fig:configuration} illustrates the corresponding configuration, where the homomorphic image of $b$ appears clearly. The next proposition summarizes the above discussion.
%
\begin{proposition}
 {\rm{(\cite{lld}).}}\label{prop:config-cond}
Let $b$ be a $2$-connected graph with vertex set $V=[n]$ and consider a function $h:V\rightarrow\Z$ and a bijection $\beta:V\rightarrow[n]$ satisfying $\beta(n)=n$.
Then the simplex $\Po(h,\beta)$ corresponding to the pair $(h,\beta)$ is contained in the polytope $\Po(\beta)$ if and only if the following condition is satisfied:
%We set $\xi_i:=\beta(i)/n$, $i=1,\dots,n$.
%Then the pair $(h,\beta)$ determines a valid subpolytope $\Po(h,\beta)$ of $\Pob$ if and only if the following condition is satisfied:
%
\begin{equation} \label{eq:config-cond}
\mathrm{for\ any\ edge}\ \{i,j\}\ \mathrm{of}\ b,\ \  \beta(i)<\beta(j) \ \ \mathrm{implies}\ \ h_i=h_j \ \mathrm{or} \ h_i=h_j+1 .
\end{equation}
\end{proposition}
%% figure
\begin{figure}[H]
\begin{center}
%\scalebox{.6}{\includegraphics{representation.eps}}
\scalebox{.75}{\input{FractionalEx2.pstex_t}}
\vspace{.5cm}
\caption{Fractional representation of a simplicial subpolytope of a graph $b$}
\label{fig:configuration}
\end{center}
\end{figure}
%%

\begin{cor}
{\rm{(\cite{lld}).}} \label{prop:VolPoc}
Let $b$ be a $2$-connected graph and let $\nu(b)$ be the number of pairs $(h,\beta)$ such that the condition \emph{(\ref{eq:config-cond})} is satisfied. Then the volume of the polytope $\Pob$ is given by
%
\begin{equation} \label{eq:VolPoc}
\Vol(\Pob) = \nu(b)/(n-1)!.
\end{equation}
%
\end{cor}

%
\medskip
Proposition~\ref{prop:VolPoc} can be used to compute the weight of some families of graphs, since $w_M(b)=(-1)^{e(b)}\Vol(\Pob)$. 

%%%%%%%%%%%%%%
% Ree--Hoover configurations
%%%%%%%%%%%%%%%%
In a similar way we can adapt the above configurations to the context of the Ree--Hoover weight.
%%%%%%%%%%%%% Prop RH %%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}
 \label{prop:RHconfig-cond}
Let $b$ be a $2$-connected graph with vertex set $V=[n]$ and consider a function $h:V\rightarrow\Z$ and a bijection $\beta:V\rightarrow[n]$ satisfying $\beta(n)=n$.
%We set $\xi_i:=\beta(i)/n$, $i=1,\dots,n$.
Then the simplex $\Po(h,\beta)$ corresponding to the pair $(h,\beta)$ is contained in the polytope $\Po_{RH}(b)$ if and only if the following conditions are satisfied:
%the pair $(h,\beta)$ determines a valid RH-subpolytope $\Po_{RH}(h,\beta)$ if and only if the following conditions are satisfied:
%
\begin{equation} \label{eq:Mconfig-cond}
 \mathrm{for\ any\ edge}\ \{i,j\}\ \mathrm{of}\ b,\ \  \beta(i)<\beta(j) \ \ \mathrm{implies}\ \ h_i=h_j \ \mathrm{or} \ h_i=h_j+1,\hspace*{.7cm}
\end{equation}
%\quad \quad \quad \quad and
\begin{equation} \label{eq:RHconfig-cond}
\mathrm{for\ any\ edge}\ \{i,j\}\ \mathrm{of}\ \overline{b},\ \  \beta(i)<\beta(j) \ \ \mathrm{implies}\ \ h_i\leq h_j-1 \ \mathrm{or} \ h_i\geq h_j+2.
\end{equation}
\end{proposition}

%
%\noindent
\begin{proof} 
%Let us denote by $\Po_{RH}(b)$ the union of all valid RH-subpolytopes $\Po_{RH}(h,\beta).$ The closure of  $\Po_{RH}(b)$ is a union of polytopes. 
%Let $X_{\can}=(x_1,\dots,x_{n})$ denote the canonical point associated
%with $(h,\beta)$, i.e. where $x_i= h_i + \xi_i$, with
%$\xi_i:=\beta(i)/n$, $i=1,\dots,n$. Then the pair RH-$(h,\beta)$
%determines a valid subpolytope $\Po_{RH}(h,\beta)$ of $\Po_{RH}(b)$ if and only
%if the point $X_{\can}$ is in $\Po_{RH}(b)$.  But, 
%The 
Condition
(\ref{eq:Mconfig-cond}) expresses exactly that $|x_i-x_j|<1$,
whenever $\{i,j\}$ is an edge of $b$, and condition
(\ref{eq:RHconfig-cond}) expresses exactly that $|x_i-x_j|>1$,
whenever $\{i,j\}$ is an edge of $\overline{b}$, that is, the defining
conditions for the domain of integration of (\ref{eq:wRH}).     \end{proof}
%%%%%%%%%%%%%%%
%\begin{proposition}
% \label{prop:RHVolsubPoc}
%Let $b$ be a $2$-connected graph and let $(h,\beta)$ be such that $\beta(n)=n$ and condition
%\emph{(\ref{eq:Mconfig-cond})} and \emph{(\ref{eq:RHconfig-cond})} are satisfied.  Then the volume of the associated subpolytope $\Po_{RH}(h,\beta)$ is equal to $1/(n-1)!$.
%\end{proposition}

%
%\noindent
%\begin{proof}
%The proof is the same as the one of Proposition~$24$ (\cite{lld}) and is left to the reader.
%\end{proof}
%
\begin{proposition}
 \label{prop:RHVolPoc}
Let $b$ be a $2$-connected graph and let $\nu_{RH}(b)$ be the number of pairs $(h,\beta)$ such that conditions \emph{(\ref{eq:Mconfig-cond})} and \emph{(\ref{eq:RHconfig-cond})} are satisfied. Then the volume of $\Po_{RH}(b)$ is given by
%
\begin{equation} \label{eq:RHVolPoc}
\Vol(\Po_{RH}(b)) = \nu_{RH}(b)/(n-1)!.
\end{equation}
%
\end{proposition}

%
%\noindent 
\begin{proof} It is clear that the
polytope $\Po_{RH}(b)$ is the disjoint union of all its subpolytopes
$\Po(h,\beta)$ and the result follows immediately.
\end{proof}
Proposition~\ref{prop:RHVolPoc} can be used to compute the weight of some families of graphs, since $w_{RH}(b)=(-1)^{e(b)}\Vol(\Po_{RH}(b))$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 3 :Ree--Hoover weight of some infinite families of graphs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Ree--Hoover weight of some infinite families of graphs} \label{sec: new}
Here are  some of our results concerning 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 \emph{graph homorphisms}.  
The weights of $2$-connected graphs $b$ are given in absolute value $|w(b)|$, the sign being always equal to $(-1)^{e(b)}$.
\begin{lem}\label{carrecroix}
Suppose that $g$ is a graph over $[n]$ and $i,j\in[n-1]$ 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 $(h,\beta)$ (with $h_n=-1,\ \beta(n)=n$) satisfies either one of the following conditions:
\begin{enumerate}
\item $h_i=1$, $h_j=0$ and $\beta(i)<\beta(j)$, 
\item $h_i=-2$, $h_j=-1$ and $\beta(i)>\beta(j)$.
\end{enumerate}
\end{lem}

\subsection{The Ree--Hoover weight of the graph $K_n\backslash S_k$}
%%%%%%%%%%%%%%    K-n\Sk                  %%%%%%%%%%%
Let $S_k$ denote the $k$-star graph with vertex set $[k+1]$ and edge set $\{\{1, 2\}, \{1, 3\},\ldots,
\{1, k+1\}\}$.
%%%%%%%%%%%%%     S1           %%%%%%%%%%%%%%%%%
As a first example, we compute $|w_{RH}(K_n\backslash e)|,$ with $K_n\backslash e=K_n\backslash S_1.$
%% prop
\begin{proposition}
\label{prop55}
  For $n\geq3$, let $K_n\backslash e$ denote the complete graph on $n$ vertices from which an arbitrary edge has been removed. Then we have
%%
%
\begin{equation} \label{eq:Knmoinse}
|w_{RH}(K_n\backslash e)|= {\frac 2 {(n-1)}}\cdot
\end{equation}
\end{proposition}

%
%proof prop 
\begin{proof}  
We can assume that the missing edge is $e=\{1,n\}$. According to Lemma~\ref{carrecroix} with $1,\ldots,k$, $j=k+1,\ldots ,n-1$, there are two possibilities for $h\,$: a) set $h_1=1$, $h_n=-1$ and all other $h_i=0$, so that
$\beta(1)$ must be 1, and b) set $h_1=-2$ and all other $h_i=-1$, so that
$\beta(1)$ must be $n-1$. In both cases $\beta$ can be extended in $(n-2)!$ ways, giving the possible relative positions of the $x_i,$ $2 \leq i \leq n-1$ (see Figure~\ref{fig:configurationS1}). So, there are $2(n-2)!$ RH-configurations $(h, \beta).$ 
%% figure
\begin{figure}[H]
\begin{center}
\scalebox{.55}{\input{configurationS1.pstex_t}}
\caption{Fractional representation of a simplicial subpolytope of $\Po_{RH}(K_n\backslash e)$}
\label{fig:configurationS1}
\end{center}
\end{figure}
%
\end{proof}
 %% end proof
 Note that, from (\ref{eq:wM=wRH}) and the known fact that $| w_M(K_n)|= n$, we have
 
% , relation (\ref{eq:2w_M et w_RH}) implies the following result 
 
 
 %
\begin{equation} \label{eq:wKnmoinse}
w_{M}(K_n\backslash e)= (-1)^{{\binom n 2}-1}\left(n+{\frac 2 {(n-1)}}\right),
\end{equation}
%
%due to Bodo Lass \cite{Lass}, 
since 
 %
\begin{equation*} \label{eq:wMKnmoinse}
|w_{M}(K_n\backslash e)|= | w_{RH}(K_n)|+|w_{RH}(K_n\backslash e)|.
\end{equation*}
%
In the general case we have:
%%%%%%%%%%%%%%    S_k  %%%%%%%%%%%%%%%
\begin{proposition}
\label{prop:Sk}
For $ k \geq1$, $ n \geq k+3$, we have
\begin{equation}
\label{eq:wRH Sk} | w_{RH}(K_n\backslash S_k)|=\frac{2k!}{(n-1)(n-2)\cdots(n-k )}\cdot
\end{equation}
\end{proposition}

% proof
\begin{proof}
We assume that the missing edges are $\{1,n\},$ $\{2,n\},$ $\ldots,$ $\{k,n\}$ (see Figure~\ref{fig:S3}, for the case of $S_3$).
%% figure
\begin{figure}[H]
\begin{center}
\scalebox{.85}{\input{S3.pstex_t}}
\caption{The graph $S_3$}
\label{fig:S3}
\end{center}
\end{figure}  
%
According to Lemma~\ref{carrecroix} with $1,\ldots,k$, $j=k+1,\ldots ,n-1$, there are two possibilities for $h\,$:
 \begin{itemize}
  \item $h_1=h_2=\cdots=h_k=1$ and $h_n=-1$ and all other $h_i=0$, so that
$(\beta(1),\beta(2),\ldots,\beta(k))$ must be  a permutation of $\{1,2,\ldots,k\},$ 
 \item $h_1=h_2=\cdots=h_k=-2$ and all other $h_i=-1$, so that
$(\beta(1),\beta(2),\ldots,\break\beta(k))$ must be  a permutation of $\{n-1,n-2,\ldots,n-k\}.$
\end{itemize}
In each case $\beta$ can be extended in $(n-(k+1))!$ ways, giving the possible relative positions of the $x_i,$ $k+1 \leq i \leq n-1$ (see Figure~\ref{fig:configurationS3}, for the case of $S_3$). So, there are $2k!(n-k-1)!$ RH-configurations $(h, \beta)$. 
%% figure
\begin{figure}[H]
\begin{center}
\scalebox{.55}{\input{configurationS3.pstex_t}}
\caption{Fractional representation of a simplicial subpolytope of $\Po_{RH}(K_n\backslash S_3)$}
\label{fig:configurationS3}
\end{center}
\end{figure}
%
\end{proof}
%%%%%%%%%%%%%%%%              Sj--Sk            %%%%%%%%%%%%%%%%%%%%
 \subsection{The Ree--Hoover weight of the graph $K_n\backslash (S_j$--$S_k)$}
 Let $S_j$--$S_k$ denote the graph obtained by joining with a new edge the centers of a $j$-star and of a $k$-star. See  Figure~\ref{lesdeux a)} for an example.
 %% figure 
\begin{figure}[H]
\vspace{.2cm}
\centerline{\includegraphics[width=.35\textwidth]{S3-S4ne.eps}}
 \caption{The graph $S_3$--$S_4$}
 \label{lesdeux a)}
 \end{figure}
  %%%%%%%%%%%%%%%   S1-S1            %%%%%%%%%%%%%%%%
  Let us start with the simple case $S_1$--$S_1.$
\begin{proposition}
\label{propS1-S1}
For $ n \geq 5 $, we have
%let $g_n=K_n\backslash S_1$--$S_1.$ Then we have
\begin{equation}
\label{ eq:wRH S1-S1}
|w_{RH}(K_n\backslash S_1\! \!-\! \!S_1)|=\frac{2}{(n-1)(n-2)(n-3)}\cdot
\end{equation}
\end{proposition}

%
\begin{proof}
We can assume that the missing edges are $\{1,n\},$ $\{2,n\}$ and $\{2,3\}$ (see Figure~\ref{fig:S1-S1}).
% figure
According to Lemma~\ref{carrecroix} with $i=1,\ldots ,k+1$ and $j=k+2,\ldots ,n-1$ for $i\neq 2$ and $j=k+j+2,\ldots ,n-1$ for $i=2$,
there are two possibilities for $h\,$:
 \begin{itemize}
  \item $h_1=h_2=1$ and $h_n=-1$ and all other $h_i=0$, so that
$(\beta(1),\beta(2),\beta(3))$ must be  $(1,3,2),$ 
 \item $h_1=h_2=-2$ and all other $h_i=-1$, so that
$(\beta(1),\beta(2),\beta(3))$ must be $(n-1,n-3,n-2).$
\end{itemize}
In each case $\beta$ can be extended in $(n-4)!$ ways, giving the possible relative positions of the $x_i,$ $4 \leq i \leq n-1$ (see Figure~\ref{fig:configurationS1-S1}). So, there are $2(n-4)!$ RH-configurations $(h, \beta)$. 
% figure
\begin{figure}[H]
\begin{center}
\scalebox{.65}{\input{S1-S1.pstex_t}}
\caption{The graph $S_1$--$S_1$}
\label{fig:S1-S1}
\end{center}
\end{figure}  
%
 % figure
\begin{figure}[H]
\begin{center}
\scalebox{.55}{\input{configurationS1-S1.pstex_t}}
\caption{Fractional representation of a simplicial subpolytope of $\Po_{RH}(K_n\backslash S_1$--$S_1)$}
\label{fig:configurationS1-S1}
\end{center}
\end{figure}
% 
\vspace{-.4cm}
\end{proof}
In the general case we have:
%%%%%%%%%%%%%%%%           Sj-SK        %%%%%%%%%%%%%%
\begin{proposition}
\label{propSj-Sk}
For $j \geq k \geq 1,$ $ n \geq k+j+3,$ we have
%let $g_n=K_n\backslash (S_j$--$ S_k).$ Then we have
\begin{equation}
 \label{eq:wRH Sj-SK } |w_{RH}(K_n\backslash (S_j \! \!-\! \!S_k))|=\frac{2k!j!}{(n-1)(n-2)\cdots(n-(k+j+1))}\cdot
 \end{equation}
\end{proposition}
% preuve 
\begin{proof}We can assume that the missing edges are $\{1,n\},$ $\{2,n\},$ $\ldots,$ $\{k+1,n\}$ and $\{2,k+2\},$ $\{2,k+3\},$ $\ldots,$$\{2,k+j+1\}$ (see Figure~\ref{fig:S2-S3}, for the case of  $S_2$--$S_3$).  
% figure
\begin{figure}[h]
\begin{center}
\scalebox{.85}{\input{S2-S3.pstex_t}}
\caption{The graph $S_2$--$S_3$}
\label{fig:S2-S3}
\end{center}
\end{figure}  
%
According to Lemma~\ref{carrecroix} there are two possibilities for $h\,$:
 \begin{itemize}
 %
\item $h_1=h_2=\cdots=h_{k+1}=1$ and $h_n=-1$ and all other $h_i=0$, so that
$(\beta(1),\beta(3),\ldots,\beta(k+1))$ must be a permutation of $\{1,2,\ldots,k\}$  and\break $(\beta(k+2),\beta(k+3),\ldots,\beta(k+j+1))$ must be a permutation of\break $\{k+1,\ldots,k+j\}$ and $\beta(2)=k+j+1.$
%
\item $h_1=h_2=\cdots=h_{k+1}=-2$ and all other $h_i=-1$, so that
$(\beta(1),\beta(3),\ldots,\break\beta(k+1))$ must be a permutation of $\{n-1,n-2,\ldots,n-k\}$  and\break $(\beta(k+2),\beta(k+3),\ldots,\beta(k+j+1))$ must be a permutation of\break $\{n-k-1,\ldots,n-k-j\}$ and $\beta(2)=n-k-j-1.$
\end{itemize}
In each case $\beta$ can be extended in $(n-(k+j+2))!$ ways, giving the possible relative positions of the $x_i,$ $k+j+2 \leq i \leq n-1$ (see Figure~\ref{fig:configurationS2-S3}, for the case of  $S_2$--$S_3$). So, there are $2k!j!(n-(k+j+2))!$ RH-configurations $(h, \beta)$. 
 % figure
\begin{figure}[H]
\begin{center}
\scalebox{.55}{\input{configurationS2-S3.pstex_t}}
\caption{Fractional representation of a simplicial subpolytope of $\Po_{RH}(K_n\backslash S_2$--$S_3)$}
\label{fig:configurationS2-S3}
\end{center}
\end{figure}
%
\end{proof}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{The Ree--Hoover weight of the graph $K_n\backslash (C_4\cdot S_k)$}
Let $C_4\cdot S_k$ denote the graph obtained by identifying one vertex of the graph $C_4$ with the center of a $k$-star. See  Figure~\ref{lesdeux b)} for an example. 
 %% figure 
\begin{figure}[ht]
\vspace{.2cm}
\centerline{\includegraphics[width=.30\textwidth]{C4.S4ne.eps}}
 \caption{The graph $C_4\cdot S_4$}
 \label{lesdeux b)}
 \end{figure}
 %
\medskip
 
 Let us start with the special case $k=0$ corresponding to the graph $K_n\backslash C_4$. The proof is diffferent from the one corresponding to the case $k \geq 1.$
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsubsection{The Ree--Hoover weight of the graph $K_n\backslash C_4$}
%%%%%%%%%%%%%        C_4             %%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}
\label{propC4}
For $n \geq 6,$ we have
%
\begin{equation}
\label{eq:wRH C4} | w_{RH}(K_n\backslash C_4)|=\frac{8}{(n-1)(n-2)(n-3 )},
\end{equation}
%
where $C_4$ is the
unoriented cycle with $4$ vertices. 
\end{proposition}

%
%
\begin{proof}
We can assume that the missing edges are $\{1,n\},$ $\{2,n\},$ $\{2,3\}$ and $\{3,1\}$ (see Figure~\ref{fig:C4}). 
There are four possibilities for $h\,$:
 \begin{itemize}
 %
  \item $h_1=h_2=1$ and $h_n=-1$ and all other $h_i=0$, so that
$(\beta(1),\beta(2))$ must be a permutation of $\{2, 3\}$ and $\beta(3)=1,$ 
%
  \item $h_1=h_2=1$ and $h_n=h_3=-1$ and all other $h_i=0$, so that
$(\beta(1),\beta(2))$ must be a permutation of $\{1, 2\}$ and $\beta(3)=n-1,$ 
%
 \item $h_1=h_2=-2$ and all other $h_i=-1$, so that
$(\beta(1),\beta(2))$ must be a permutation of $(n-2,n-3)$ and $\beta(3)=n-1,$ 
%
 \item $h_1=h_2=-2,$ $h_3=0$ and all other $h_i=-1$, so that
$(\beta(1),\beta(2))$ must be a permutation of $(n-1,n-2)$ and $\beta(3)=1.$ 
\end{itemize}
In each case $\beta$ can be extended in $(n-4)!$ ways, giving the possible relative positions of the $x_i,$ $4 \leq i \leq n-1$ (see Figure~\ref{fig:configurationC4}). So, there are $4\cdot2!(n-4)!$ RH-configurations $(h, \beta)$. 
% figure
\begin{figure}[H]
\begin{center}
\scalebox{.75}{\input{C4.pstex_t}}
\caption{The graph $C_4$}
\label{fig:C4}
\vspace{-.15cm}
\end{center}
\end{figure}    
%
% figure 
\begin{figure}[H]
\begin{center}
\scalebox{.55}{\input{configurationC4.pstex_t}}
\caption{Fractional representation of a simplicial subpolytope of $\Po_{RH}(K_n\backslash C_4)$}
\label{fig:configurationC4}
\end{center}
\end{figure}
%
\vspace{-2.2cm}
\end{proof}
%\vspace{-1.9045cm}
%%%%%%%%%%%%%%%        C_4 -Sk           %%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}
\label{prop:C4.Sk}
For $k \geq 1,$ $ n \geq k+5,$ we have
\vspace{-1.4045cm}
\begin{equation}\label{eq:wRH C4.Sk}
%\vspace{-1.4045cm}
 | w_{RH}(K_n\backslash
(C_4\cdot S_k))|=\frac{4k!}{(n-1)(n-2)\cdots(n-(k +3))}.
\end{equation}
\end{proposition}

% proof
\begin{proof} We can assume that the missing edges are $\{1,n\},$ $\{2,n\},$ $\{4,n\},$ $\ldots,\break\{k+3,n\}$ and $\{1,3\},$ $\{2,3\}$ (see Figure~\ref{fig:C4 .S2}, for the case of $C_4\cdot S_2$).  
% figure 
\begin{figure}[H]
\begin{center}
\scalebox{.75}{\input{C4.S2.pstex_t}}
\caption{The graph $C_4\cdot S_2$}
\label{fig:C4 .S2}
\vspace{-.05cm}
\end{center}
\end{figure}  
%
\vspace{-.2cm}
According to Lemma~\ref{carrecroix} there are two possibilities for $h\,$:
 \begin{itemize}
 %
 \item $h_1=h_2=h_4\cdots=h_{k+3}=1$ and $h_n=-1$ and all other $h_i=0$, so that
$(\beta(4),\beta(5),\ldots,\beta(k+3))$ must be a permutation of $\{1,2,\ldots,k\}$  and $(\beta(1),\beta(2))$ must be a permutation of $\{k+2, k+3\}$ and $\beta(3)=k+1,$
%
\item $h_1=h_2=h_4\cdots=h_{k+3}=-2$ and all other $h_i=-1$, so that
$(\beta(4),\beta(5),\break\ldots,\beta(k+3))$ must be a permutation of $\{n-1,n-2,\ldots,n-k\}$  and $(\beta(1),\beta(2))$ must be a permutation of $\{n-k-2, n-k-3\}$ and $\beta(3)=n-k-1.$
\end{itemize}
In each case $\beta$ can be extended in $(n-(k+4))!$ ways, giving the possible relative positions of the $x_i,$ $k+4 \leq i \leq n-1$ (see Figure~\ref{fig:configurationC4 .S2}, for the case of $C_4\cdot S_2$). So, there are $2\cdot 2!\,k!\,(n-(k+4))!$ RH-configurations $(h, \beta)$. 
 % figure
\begin{figure}[H]
\begin{center}
\scalebox{.55}{\input{configurationC4-S2.pstex_t}}
\caption{Fractional representation of a simplicial subpolytope of $\Po_{RH}(K_n\backslash( C_4\cdot S_2))$}
\label{fig:configurationC4 .S2}
\end{center}
\end{figure}
\end{proof}
Note that the formula (\ref{eq:wRH C4}) is not a special case of (\ref{eq:wRH C4.Sk}).
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{The Ree--Hoover weight of the graph $K_n\backslash P_k$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%     the path graph      %%%%%%%%%
Let $P_k$ denote the path graph with vertex set
$[k]$ and edge set $\{\{1, 2\}, \{2, 3\},\ldots, \{k-1, k\}\}$.
%
  For $n \geq 5,$ we have $K_n\backslash P_3=K_n\backslash S_2$ and
  $K_n\backslash P_4=K_n\backslash S_1$--$S_1.$ Thus, the weights $w_{RH}(K_n\backslash P_3)$ and $w_{RH}(K_n\backslash P_4)$ can be obtained as special cases of Propositions~\ref{prop:Sk} and \ref{propS1-S1}. Moreover, $w_{RH}(K_n\backslash P_k)=0,$ for $k \geq 5,$ $n \geq k+1,$ which is a consequence of Theorem~\ref{theo:wRH=0}. Indeed, $C_4 \ \overline{\subseteq} \  K_n \backslash P_k$ and we conclude using (\ref{eq1:wRH=0}).
 % \\
  % rem
%  Note that the following phenomenon can occur for some $2$-connected graphs: Suppose that  $K_n\backslash g$ is $2$-connected for every $n \geq p$ and that there exists a formula $f(n)$ such that
  %
%  \begin{equation}
%\label{ }
%w_{RH}(K_n\backslash g)=f(n) \quad {\rm for} \quad n>p,
%\end{equation}
%  %
%  then for $n=p,$ we can have $w_{RH}(K_p\backslash g)=mf(p),$ $1< m \in \N.$\\
  %This is the case, for example, for the graph $g=$ for which $p=$ and $m=$.
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 3 :Ree--Hoover weight of some infinite families of graphs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Mayer weight of some infinite families of graphs} \label{sec: newMayer}
In this section, we give explicit formulas for the Mayer weight of the above infinite families of graphs. In order to do so, we use the following formula
\begin{equation}\label{eq:rondcroix}
|w_M(b)|=\sum_{b\subseteq d\subseteq K_n}|w_{RH}(d)|
\end{equation}
which is a consequence of (\ref{eq: w_M  et w_RH}) since $|w_M(b)|=(-1)^{e(b)}w_M(b)$ and $|w_{RH}(d)|=(-1)^{e(d)}w_{RH}(d)$ in the case of hard-core continuum gases in one dimension. Substituting $K_n\backslash g$ and $K_n\backslash k$ for $b$ and $d$ in (\ref{eq:rondcroix}), we have
%In this case, the computation of the Mayer weight is more difficult. Instead of using the method of graph homomorphisms, we use Formula (\ref{eq:2w_M et w_RH}) and Theorem~\ref{theo:wRH=0}. Indeed, for a graph $b$, we collect in the corresponding sum in (\ref{eq:2w_M et w_RH}), only the non-zero Ree--Hoover weights of graphs $d$ with $b\subseteq d\subseteq K_n$. These graphs are classified up to isomorphism and their multiplicities are computed in each case. Since the graphs are of the form $K_n \backslash g$, the over graphs of $K_n \backslash g$ are of the form $K_n \backslash h$, where $h$ is a subgraph of $g,$ $h \subseteq g.$ Using Formula (\ref{eq:2w_M et w_RH}) we have
%
\begin{align}
\notag 
|w_{M}(K_n \backslash g)| &=\sum_{k\subseteq g}|w_{RH}(K_n \backslash k)| \\
&=\sum_{\widetilde{h}\subseteq\widetilde{g}}m(\widetilde{h},\widetilde{g}) |w_{RH}(K_n \backslash h)|,
\end{align}
%
%where $\Omega (g)$ is a system of representatives of the unlabelled subgraphs of $g$ and $m(h)$ is the multiplicity of the graph $h.$ 
where $\widetilde{g}$ denotes the unlabelled graph corresponding to $g$, $\widetilde{h}$ runs through the unlabelled subgraphs of $\widetilde{g}$ and $m(\widetilde{h},\widetilde{g})$ is the number of ways of obtaining $\widetilde{h}$ by removing some edges in $\widetilde{h}$. In the following propositions these multiplicities $m(\widetilde{h},\widetilde{g})$ are obtainable in each case by direct combinatorial arguments.
%%%%%%%%%%%%%  Sk  %%%%%%%%%%%%%%%
\subsection{The Mayer weight of the graph $K_n\backslash S_k$}
% prop 
\begin{proposition}
\label{propM:Sk}
For $ k \geq 1 $, $ n \geq k+3 $, we have
%
\begin{equation*}
  | w_M(K_n\backslash S_k)|=n+2 \sum_{j=1}^{k} \frac
{j!\binom{k}{j}}{(n-1)(n-2)\cdots (n-j )}.
\end{equation*}
%
\end{proposition}

% proof
\begin{proof}
The over graphs of $K_n\backslash S_k$ are up to isomorphism of the form: $K_n\backslash S_j$, $1\leq j \leq k$ and $K_n.$ Their multiplicities $m(S_j,S_k)$ are given by $m(S_j,S_k)={\binom k j}$ because one has to choose $j$ edges among the $k$ edges incident to the center of the star. Note that $S_0=\emptyset$ is the star with no edge.  Hence,
%
\begin{equation*}
|w_{M}(K_n\backslash S_k)|=\sum_{j=0}^{k} \binom{k}{j}|w_{RH}(K_n\backslash S_j)|.
\label{eq:proof Sk}
\end{equation*}
We conclude using  Proposition~\ref{prop:Sk}.
%
\end{proof}
%%%%%%%%%%%%%%%%           Sj-SK        %%%%%%%%%%%%%%
\subsection{The Mayer weight of the graph $K_n\backslash S_j$--$ S_k$}
\begin{proposition}
%\label{propSj-Sk}
For $j \geq k \geq 1,$ $ n \geq k+j+3,$ we have, with the usual convention ${\binom {k+1} \ell}=0$ if $\ell>k+1$,
%let $g_n=K_n\backslash (S_j$--$ S_k).$ Then we have
%
\begin{multline*}\label{eq:wM Sj-Sk}
 \notag | w_M(K_n\backslash (S_j \! \!-\! \!S_k))| =  n+\sum_{l=1}^{j+1}2\left[\binom{j+1}{l}+\binom{k+1}{l}\right] \frac{l!}{\left(n-1\right)\cdots\left(n-l\right)}\\
 +  \sum_{m=1}^{j}\sum_{l=1}^{k}2\binom{j}{m}\binom{k}{l}\frac{m!l!}{\left(n-1\right)\cdots\left(n-(m+l+1)\right)}-\frac{2}{\left(n-1\right)}.
 \end{multline*}
 %
\end{proposition}

% proof
\begin{proof}
The over graphs of $K_n\backslash S_j$--$ S_k$ whose Ree--Hoover weight is not zero are up to isomorphism of the form: $K_n\backslash S_l$, $1\leq l \leq j+1,$ $K_n\backslash(S_m$--$S_l),$ $1\leq m \leq j,$ $1\leq l \leq k$ and $K_n.$ Their multiplicities are given by
\begin{align*}
&\textstyle
m(S_l,S_j \! \!-\! \!S_k)=\binom{j+1}{l}+\binom{k+1}{l},\ l\geq 2,\ \quad m(S_1,S_j \! \!-\! \!S_k)=\binom{j+1}{l}+\binom{k+1}{l}-1,\\
&\textstyle
m(S_m \! \!-\! \!S_l,S_j \! \!-\! \!S_k)=\binom{j}{m} \binom{k}{l}.
\end{align*}
Hence,
%
\begin{multline*}
|w_{M}(K_n\backslash (S_j \! \!-\! \!S_k))|= |w_{RH}(K_n)|+\sum_{l=1}^{j+1}\left[ \binom{j+1}{l}+\binom{k+1}{l}\right] |w_{RH}(K_n\backslash S_l)|\\
  +  \sum_{m=1}^{j} \sum_{l=1}^{k} \binom{j}{m} \binom{k}{l}|w_{RH}(K_n\backslash (S_m\! \!-\! \!S_l))|-|w_{RH}(K_n\backslash S_1)|.
\end{multline*}
We conclude using Propositions~\ref{prop:Sk} and \ref{propSj-Sk}.
%
\end{proof}
%%%%%%%%%%%%%        C_4             %%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Mayer weight of the graph $K_n\backslash C_4$}
\begin{proposition}
%\label{propC4}
For $n \geq 6,$ we have
%
\begin{equation*}
 | w_M(K_n\backslash C_4)|=n+\frac{8}{(n-1)}
+\frac{16}{(n-1)(n-2)}+\frac{16}{(n-1)(n-2)(n-3)}.
\end{equation*}
%
\end{proposition}

% proof
\begin{proof}
%The over graphs of $K_n\backslash C_4$ whose Ree--Hoover weight is not zero are up to isomorphism of the form: $K_n\backslash S_1$, $K_n\backslash S_2,$ $K_n\backslash C_4,$ $K_n\backslash(S_1$--$S_1)$ and $K_n.$ 
Note first that the RH-weight of the over graph $K_n\backslash S_1\backslash S_1$ of $K_n\backslash C_4$ is 0. The other over graphs of $K_n\backslash C_4$ are, up to isomorphism, of the form: $K_n\backslash S_1$, $K_n\backslash S_2$, $K_n\backslash C_4$, $K_n\backslash (S_1-S_1)$ and $K_n$. Their multiplicities are given by 
$$m(S_l,C_4)=4,\ 1\leq l \leq 2,\ \quad m(S_1 \! \!-\! \!S_1,C_4)=4.$$
Hence,
%
\begin{multline*}
\notag |w_{M}(K_n\backslash C_4)|= |w_{RH}(K_n)|+\sum_{l=1}^{2}4 |w_{RH}(K_n\backslash S_l)|\\
  +4|w_{RH}(K_n\backslash (S_1-S_1))|+|w_{RH}(K_n\backslash C_4)|.
\label{eq:proof C4}
\end{multline*}
We conclude using Propositions~\ref{prop:Sk}, \ref{propS1-S1} and \ref{propC4}.
%
\end{proof}
%%%%%%%%%%%%%%%        C_4 -Sk           %%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Mayer weight of the graph $K_n\backslash (C_4\cdot S_k)$}
%
\begin{proposition}
%\label{prop:C4.Sk}
For $k \geq 1,$ $ n \geq k+5,$ we have
%let $g_n=K_n\backslash(C_4\cdot S_k).$ Then we have
%
\begin{align*} 
\notag | w_{M}(K_n\backslash
(C_4\cdot S_k))|& =  n+\sum_{l=1}^{k+2}2\binom{k+2}{l}\frac{l!}{\left(n-1\right)\cdots\left(n-l\right)}\\
%
\notag & \quad \quad   +\sum_{l=1}^{k}4\binom{k}{l}\frac{l!}{\left(n-1\right)\cdots\left(n-\left(l+3\right)\right)}\\
& \quad \quad   +\sum_{l=1}^{k+1}4\left[\binom{k}{l-1}+\binom{k}{l}\right]\frac{l!}{\left(n-1\right)\cdots\left(n-\left(l+2\right)\right)}\\
\notag & \quad \quad   +\frac{4}{n-1}+\frac{12}{\left(n-1\right)\left(n-2\right)}+\frac{16}{\left(n-1\right)\left(n-2\right)\left(n-3\right)}.
\end{align*}
%
\end{proposition}

% proof
\begin{proof}
The over graphs of $K_n\backslash (C_4\cdot S_k)$ whose Ree--Hoover weight is not zero are up to isomorphism of the form: $K_n\backslash S_l$, $1\leq l \leq k+2,$ $K_n\backslash (C_4\cdot S_l),$ $1\leq l \leq k,$ $K_n\backslash(S_1$--$S_l),$ $1\leq l \leq k+1,$ $C_4$ and $K_n.$ Their multiplicities are given by 
\begin{align*}
&\textstyle
m(S_l,C_4\cdot S_k)=\binom{k+2}{l},\ l \geq 3,\ \quad m(S_1,C_4\cdot S_k)=\binom{k+2}{l}+2, \\ 
&\textstyle
m(S_2,C_4\cdot S_k)=\binom{k+2}{l}+3,\quad
m(C_4\cdot S_l,C_4\cdot S_k)=\binom{k}{l}, \\
&\textstyle
m(S_1 \! \!-\! \!S_l,C_4\cdot S_k)=2 \left[ \binom{k}{l-1} + \binom{k}{l}\right],\ l \geq 2,\quad m(S_1 \! \!-\! \!S_1,C_4\cdot S_k)=2  \binom{k}{1} +4.
\end{align*}
Hence,
%
\begin{align*}
\notag |w_{M}(C_4\cdot S_k)|&= |w_{RH}(K_n)|+\sum_{l=1}^{k+2}\binom{k+2}{l} |w_{RH}(K_n\backslash S_l)|\\
%
&\quad \quad +\sum_{l=1}^{k}\binom{k}{l} |w_{RH}(K_n\backslash C_4 \cdot S_l)|+2|w_{RH}(K_n\backslash S_1)|\\
\notag  &  \quad \quad  +\sum_{l=1}^{k+1} 2 \left[ \binom{k}{l-1} + \binom{k}{l}\right] |w_{RH}(K_n\backslash (S_1-S_l))|\\
& \quad \quad +3|w_{RH}(K_n\backslash S_2)| +|w_{RH}(K_n\backslash C_4)|+4 |w_{RH}(K_n\backslash S_1-S_1)|.
%\label{eq:proof C4.Sk}
\end{align*}
We conclude using Propositions~\ref{prop:Sk}--\ref{prop:C4.Sk}.
%
\end{proof}
%%%%%%%%%%%%%       P_K            %%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Mayer weight of the graph $K_n\backslash P_k$}
\begin{proposition}
\label{propP5}
For $k \geq 5,$ $n \geq k+1,$ we have
%
\begin{equation*}
 | w_M(K_n\backslash P_k)|=n+\frac{2(k-1)}{(n-1)}
+\frac{4(k-2)}{(n-1)(n-2)}+\frac{2(k-3)}{(n-1)(n-2)(n-3)}.
\end{equation*}
%
\end{proposition}

% proof
\begin{proof}
The over graphs of $K_n\backslash P_k,$ $k \geq 5,$  whose Ree--Hoover weight is not zero are up to isomorphism of the form: $K_n\backslash S_1$, $K_n\backslash S_2,$ $K_n\backslash P_4$ and $K_n.$ Their multiplicities are given by
$$m(S_l,P_k)=k-l,\ 1 \leq l\leq 2,\ \quad m(P_4,P_k)=k-3.$$
%
Hence,
\begin{multline*}
\notag |w_{M}(K_n\backslash P_k)|= |w_{RH}(K_n)|+ (k-1)|w_{RH}(K_n\backslash S_1)|\\
  +(k-2)|w_{RH}(K_n\backslash S_2)|+(k-3)|w_{RH}(K_n\backslash P_4)|.
\end{multline*} 
We conclude using Propositions~\ref{prop:Sk} and \ref{propS1-S1}.
%
\end{proof}
%\vspace{-1cm}
%\newpage
%%%%%%%%%%%%%%%%%%%      bib            %%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%      bib            %%%%%%%%%%%%%%%%%%%%
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\end{document}
