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%\received{15 November 2007}
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\begin{document}
%
\newbox\Adr
\setbox\Adr\vbox{
\centerline{\small AMEL KAOUCHE}
\vskip18pt
\centerline{\small Universit\'e de Moncton, Campus d'Edmudston, Canada}
}

\title{Graph weights arising from statistical mechanics}
\author[Amel Kaouche]{\box\Adr}

\address{Universit\'e de Moncton, Campus d'Edmudston, Canada}

\date{}

\begin{abstract}
In statistical mechanics it is well known that the coefficients of the virial expansion for a non-ideal gas are computed using the Mayer weight of $2$-connected graphs.
We study 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 $c$. Our results are new formulas of Mayer weights and Ree--Hoover weights for special infinite families of $2$-connected graphs.
\bigskip

\noindent  {{\sc R\'esum\'e.  }}
En m\'ecanique statistique, il est bien connu que les coefficients du d\'eveloppement du viriel  pour un gaz non-idéal sont calcul\'es en utilisant le poids de Mayer des graphes $2$-connexes.
Nous \'etudions le second poids de Mayer $w_M(c)$ et de Ree--Hoover $w_{RH}(c)$ d'un graphe $2$-connexe $c$ dans le cas d'un gaz \`a noyaux durs en dimension un. Ces poids sont calcul\'es \`a partir des volumes sign\'es de polytopes convexes associ\'es naturellement au graphe $c$. Nous donnons des nouvelles formules pour le poids de Mayer et de Ree--Hoover pour des familles sp\'eciales infinies de graphes $2$-connexes.
\end{abstract}

\maketitle


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\markright{\its S\'eminaire Lotharingien de
Combinatoire \bfs 79 \rms (2020), Article~B79e\hfill}
\def\thepage{}

\section{Introduction}
In the present paper, we study \emph{Graph weights}
in the context of a non-ideal gas in a vessel $V \subseteq \R^d$. In this case, 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 (see \cite{Kaouche, KP1, KP2, lld})
%
\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 a real-valued function associated with the pairwise interaction potential of the particles,  see \cite{FordUhlenbeck,lld}. 
%For example, in the case of a Gaussian interaction, where $f(r)=-\exp(-\alpha r^2), \ \alpha>0,$ we have,  for a connected graph $c$ with $n$ vertices and $e(c)$ edges,
%$ w(c) = (-1)^{e(c)}\left( \frac{\pi}{\alpha} \right)^{\frac{d(n-1)}{2}} \gamma(c)^{-\frac{d}{2}},$
%%
%where $\gamma(c)$ is the graph complexity, which, in the case of a connected graph, is the number of spanning trees.

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}

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\markboth{\SMALL AMEL KAOUCHE}{\SMALL GRAPH WEIGHTS ARISING FROM
STATISTICAL MECHANICS}


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}). 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 $\mathcal{B}[n]$ denotes 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).

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 (\ref{eq:fftilde}), in dimension $d=1$. The values $w_{M}(c)$ and $w_{RH}(c)$ for all
$2$-connected graphs $c$ of size at most 8 are given in \cite{Kaouche,KLtables}.
%
%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 $d=1$ (see \cite{Kaouche}, \cite{KLtables}-\cite{cite{Mayer}). 
%%(see \cite{Kaouche,KLtables,KP2, KLabelle2, lld,PLep}). 
%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 new explicit formulas for the Mayer and Ree--Hoover weights of certain infinite families of graphs.
%In Section $1$
In Section~\ref{relation}, we 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}. The interest of using the Ree--Hoover weight is that it has the value zero for many graphs. 
%Section $2$
Section~\ref{Hard-core continuum} 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.
A decomposition of the polytope $\mathcal{P}(c)$ into a certain number of simplices is exploited. This method was introduced in \cite{lld} and was adapted in \cite{Kaouche,KP2} to the context of Ree--Hoover weights and is called the method of graph homomorphisms. 
%
The explicit computation of Mayer or Ree--Hoover weights of particular graphs is very difficult in general and has only been made for certain specific families of graphs (see \cite{KLabelle,KP1,KP2,KLabelle2, KLabelle3,lld}). In the present paper we extend this list to other graphs. We give new explicit formulas of the Ree--Hoover weight of these graphs in Section~\ref{sec: new}. Section~\ref{sec: newMayer} 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 by $e(g)$. If $e$ is an edge of $g$ (i.e., $e\in g$), $g\backslash e$ denote 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 $[n]=\{1,2,\ldots , n  \}$ is denoted by $K_n$. The complementary graph of a subgraph $g\subseteq K_n$ is the graph $\overline{g}=K_n\backslash g$.
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% %%%       \section   %%%%%%%%%%%%
\section{Relations between Mayer weight and Ree--Hoover weight} \label{relation}
%%%%%%%%%%%%%
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 weight series, vertices in the resulting 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{eqnarray}
%\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{eqnarray}
%%
In general, using M\"obius inversion, it is easy to state formulas linking the two weights $w_M$ and $w_{RH}.$ These formulas are implicit in \cite{RH2}.
%% porp relations entre wM et wRH
\begin{proposition}[\cite{KP2}] \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}
%
\end{proposition}
%%% preuve 
\begin{proof} See \cite{KP2}.
\end{proof}
%%%%% debut ajouter
Consequently, the virial coefficient can be rewritten in the form
\begin{equation}\label{eq:exp RH}
\beta_n=\frac{1-n}{n!}\sum_{b \in \mathcal{B}[n]}{a}_n(b)w_{RH}(b),
\end{equation}
for appropriate coefficients ${a}_n(b)$ called the \emph{star content} of the graph $b.$ The importance of (\ref {eq:exp RH}) is due to the fact that ${a}_n(b)=0$ or $w_{RH}(b)=0$ for many graphs $b.$ This greatly simplifies the computation of $\beta_n.$
%%% fin ajouter

Using the definition of the Ree--Hoover weight, we have 
%
\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}
%{\rm{(\cite{KP2}).}}
%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}
%{\rm{(\cite{KP2}).}}
%$|\mathcal{B}[n]|_{w_{M}},$ 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}
%%%% preuve 
%\begin{proof} See \rm{\cite{KP2}}.
%\end{proof}
%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{eqnarray}
% |\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{eqnarray}
%%
%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{eqnarray}
%\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{eqnarray}
%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. 
%
%% preuve
%\begin{proof}
%We have
%\begin{eqnarray*}
%\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{eqnarray*}
%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{eqnarray*}
%\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{eqnarray*}
%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)=\left\{
%\begin{array}{ll}
% g\backslash \{1,n\}, & \mathrm{if} \hspace*{0.1 cm}  \{1,n\} \in g, \\
%g\cup \{1,n\}, &  \mathrm{otherwise}. \\
%\end{array}
%\right.   \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\cdot
%\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)=\left\{
%\begin{array}{ll}
% g\backslash \{a,n\}, & \mathrm{if} \hspace*{0.1 cm}  \{a,n\} \in g, \\
%g\cup \{a,n\}, &  \mathrm{otherwise}. \\
%\end{array}
%\right.   \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} (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{eqnarray*}
%{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{eqnarray*}
%%
% \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} \label{Hard-core continuum}
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's 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 \text{ for all }
\{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{multline*}
%\label{eq17} 
\mathcal{P}_{RH}(c)=\{ X \in \mathbb{R}^{n}: x_ n =0
,\ |x_{i}-x_{j}|< 1 \text{ for all }
\{i,j\}\in c, \\
%
|x_{i}-x_{j}|> 1 \text{ for all } \{i,j\}\in \overline{c}\} \subseteq \R^{n-1} \times \{0\}\subseteq \R^n.
\end{multline*}
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   Why many graphs have Ree--Hoover weight equal to $0$ %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Why many graphs have Ree--Hoover weight equal to $0$}
\subsection{Sufficient conditions for $w_{RH}=0$}
%
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 guarantee the vanishing of their Ree--Hoover weights (see \cite{Kaouche}). We introduce first some variants of the notion of subgraph and then state 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}is a real
%\end{figure} 
%% fin fig
%$\bullet$ Case 2: if $n=2k$ (see Figure \ref{fig:C8} with $n=8$), we can take the cycle
%$$\{\{1,3\},\{3,5\},\{5,7\},\{7,2\},\{2,2k\},\{2k,2(k-1)\},\ldots,\{6,4\},\{4,1\}\}.$$non-ideal
%%% 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}[\cite{KP2}] \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 imply} \quad w_{RH}(g)=0.
\end{equation}
%
\end{proposition}
%%% preuve 
\begin{proof} See \cite{KP2}.
\end{proof}
\begin{theorem}[\cite{KP2}] \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{gather}
\label{eq1:wRH=0}
g\ \text{is chordal}:\ C_k \ \overline{\subseteq} \  g, \ \ k \geq 4,\\
\label{eq2:wRH=0}
{\rm or}\ g\ \text{is claw-free}:\ S_3 \ \overline{\subseteq} \  g,
\end{gather} 
where $S_3$ is the $3$-star graph {\rm{(}}see Figure {\rm{\ref{fig1:S3})}} and $C_k$ is the cycle on $k$ elements.
\end{theorem}
%(see Figure~\ref{fig:S3})
%% figure
\begin{figure}[H]
\begin{center}
\scalebox{.85}{\input{S3.pstex_t}}
\caption{The graph $S_3$}
\label{fig1:S3}
\end{center}
\end{figure}  
%%% preuve 
\begin{proof} See \cite{KP2}.
\end{proof}
%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% %%%       \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 $\Vol(\Pob) = \nu(b)/(n-1)!$.
 \subsection{Graph homomorphisms} 
 %
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 gas in one dimension and was adapted by Kaouche and Leroux \cite{Kaouche,KP2} to the context of Ree--Hoover weights. 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 with $b$. 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 with 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, with 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 zero 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 $h:V\rightarrow\Z$ denotes 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 configuration space $\R^{n-1}\times \{0\}$.

Using the fractional coordinates to represent the centre of gravity $X_{h,\beta}$ of the simplex $\Po(h,\beta)$, and drawing a line segment connecting $x_i=(h_i,\xi_i)$ 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 $b$ which characterizes the subpolytope $\Po(h,\beta)$. For example, take $n=6$ and $b=\{ \{1, 3\},\{1, 5\},\{1, 6\},\{2, 3\},\{2, 4\},\break\{5, 6\} \}$. 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}[\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}
\text{for any edge}\ \{i,j\}\ \text{of}\ b,\ \  \beta(i)<\beta(j) \ \ \text{implies}\ \ h_i=h_j \ \text{or} \ h_i=h_j+1 .
\end{equation}
\end{proposition}
%% figure
\begin{figure}[H]
\begin{center}
%\scalebox{.6}{\includegraphics{representation.eps}}
\scalebox{.7}{\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}[\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 fashion we can adapt the above configurations to the context of the Ree--Hoover weight.
%%%%%%%%%%%%% Prop RH %%%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}[\cite{KP2}] \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}
%%% preuve 
\begin{proof} See \cite{KP2}.
\end{proof}
%%cite{KP2}
%
%\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}[\cite{KP2}] \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}
%%% preuve 
\begin{proof} See \cite{KP2}.
\end{proof}
%%cite{KP2}
%
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 new explicit formulas for the Ree--Hoover weight of certain infinite families of
graphs. These were first conjectured from numerical values using Ehrhart polynomials.  Their proofs use the techniques of \emph{graph homomorphism} and Theorem~\ref{theo:wRH=0}.  
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 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 $K_n\backslash C_k$}
 Let  $C_k$ denote the cycle on $k$ elements.
% %
   For the special case $k=4$ corresponding to the graph $K_n\backslash C_4$ the proof is different from the one corresponding to the case $k \geq 5.$
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsubsection{The Ree--Hoover weight of the graph $K_n\backslash C_4$}
%%%%%%%%%%%%%        C_4             %%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}[\cite{KP2}] \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 \rm{(}see Figure \ref{fig:C4}). 
\end{proposition}
%%%% figure%%%
%\begin{figure}[H]
%\begin{center}
%\scalebox{.55}{\input{C4.pstex_t}}
%\caption{The graph $C_4$}
%\label{fig:C4}
%\vspace{-.15cm}
%\end{center}
%\end{figure}    
%%%
%
%
%% figure 
\begin{figure}[ht]
\vspace{.2cm}
\centerline{\includegraphics[width=.26\textwidth]{C4ne.eps}}
\caption{The graph $C_4$}
\label{fig:C4}
 \end{figure}
 %

%\begin{proof} See {\rm{(\cite{KP2}).}}
%\end{proof}
%%%%%proposition
\begin{proposition}
\label{prop:Ck}
%
For $k \geq 5,$ $ n \geq k+1,$ we have
\begin{equation}\label{eq:wRH Ck}
w_{RH}(K_n\backslash C_k) = 0.
\end{equation}
\end{proposition}

% proof
\begin{proof}
Equation (\ref{eq:wRH Ck}) is a consequence of Theorem~\ref{theo:wRH=0}.
 Indeed, $C_4 \ \overline{\subseteq} \  K_n \backslash C_k,$ for $k \geq 5,$ $ n \geq k+1$ and we conclude using (\ref{eq1:wRH=0}).
\end{proof}
 %%%%%%%%%%%%%%%%        S_j C_4 S_k      %%%%%%%%%%%%%%%%%%%
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{The Ree--Hoover weight of the graph $K_n\backslash (S_j\cdot C_4\cdot S_k)$}
 Let  $(S_j\cdot C_4\cdot S_k)$ denote the graph obtained by joining with an edge of the graph $C_4$ the centres of a $j$-star and of a $k$-star. See  Figure \ref{S3C4S4} for an example. 
 
\medskip
 %% figure 
\begin{figure}[ht]
\vspace{.2cm}
 \centerline{\scalebox{.80}{\input{S3.C4.S4ne.pstex_t}}}
 \caption{The graph $S_3\cdot C_4\cdot S_4$}
 \label{S3C4S4}
 \end{figure}
 %
 
\medskip
 %%%%%%%%%%%%%%%%  S_2. C_4.S_1   %%%%%%%%%%%%%%%%
 Let us start with the simple case $S_2 \cdot C_4\cdot S_1$.
 \begin{proposition}
\label{prop:S2.C4.S1}
For $ n \geq 8,$ we have
\begin{equation}\label{eq:wRH S2.C4.S1}
|w_{RH}(K_n\backslash (S_2\cdot C_4\cdot S_1))| = \frac{4}{(n-1)(n-2)\cdots(n-6)}.
\end{equation}
\end{proposition}

% proof
\begin{proof} We can assume that the missing edges are $\{1,n\},$ $\{2,n\},$ $\{4,n\},$ $\{1,3\},$ $\{1,5\},$ $\{1,7\}$ and $\{2,3\}$ (see Figure \ref{fig:S2.C4.S1}).  
% figure 
\begin{figure}[H]
\begin{center}
\scalebox{.55}{\input{S2.C4.S1.pstex_t}}
\caption{The graph $S_2 \cdot C_4\cdot S_1$}
\label{fig:S2.C4.S1}
\vspace{-.05cm}
\end{center}
\end{figure}  
%figure 
\vspace{-.2cm}
%
\vspace{-.2cm}
According to Lemma~\ref{carrecroix} there are two possibilities for $h$:
 \begin{itemize}
 %
 \item $h_1=h_2=h_4=1$ and $h_n=-1$ and all other $h_i=0$, 
 so that $(\beta(1),\beta(2),\break \beta(3),\beta(4))$ must be $(6,3,2,1)$
and $(\beta(5),\beta(7))$ must be a permutation of $\{4,5\},$
%
\item $h_1=h_2=h_4=-2$ and all other $h_i=-1$, so that
$(\beta(1),\beta(2),\beta(3),\beta(4))$ must be $(n-6,n-3,n-2,n-1)$
and $(\beta(5),\beta(7))$ must be a permutation of $\{n-4,n-5\}.$
\end{itemize}
In each case $\beta$ can be extended in $(n-7)!$ ways, giving the possible relative positions of the  $(n-7)$ $x_i$ (see Figure \ref{fig:configurationS2.C4 .S1}). So, there are $2\cdot 2!\,(n-7)!$ RH-configurations $(h, \beta).$
% figure
\begin{figure}[H]
\begin{center}
\scalebox{.53}{\input{configurationS2.C4.S1.pstex_t}}
\caption{Fractional representation of a simplicial subpolytope of $\Po_{RH}(K_n\backslash( S_2\cdot C_4\cdot S_1))$}
\label{fig:configurationS2.C4 .S1}
\end{center}
\end{figure}
%%%%%%%%%
\end{proof}
%%%%%%%%%%%%%%%        Sj.C_4.Sk           %%%%%%%%%%%%%%%%%%%%%%%%
In the general case we have the following result.
\begin{proposition}
\label{prop:Sj.C4.Sk}
For $j \geq k \geq 1,$ $ n \geq k+j+5,$ we have
\begin{equation}\label{eq:wRH Sj.C4.Sk}
|w_{RH}(K_n\backslash (S_j\cdot C_4\cdot S_k))| = \frac{2k!\,j!}{(n-1)(n-2)\cdots(n-(k+j+3))}.
\end{equation}
\end{proposition}

% proof
\begin{proof} We can assume that the missing edges are $\{2,n\},$ $\{4,n\},$ $\{6,n\},$ $\ldots,$ $\{2k+2,n\}$ and $\{1,3\},$ $\{1,5\},$ $\{1,7\},$ $\ldots,$ $\{2j+3,n\}$ and $\{1,n\},$ $\{2,3\}$ (see Figure \ref{fig:S2.C4.S2}, for the case of $S_2\cdot C_4\cdot S_2$).  
% figure 
\begin{figure}[H]
\begin{center}
\scalebox{.55}{\input{S2.C4.S2.pstex_t}}
\caption{The graph $S_2 \cdot C_4\cdot S_2$}
\label{fig:S2.C4.S2}
\vspace{-.05cm}
\end{center}
\end{figure}  
%
\vspace{-.2cm}
%
\vspace{-.2cm}
According to Lemma~\ref{carrecroix} there are two possibilities for $h$:
 \begin{itemize}
 %
 \item $h_1=h_2=h_4\cdots=h_{2k+2}=1$ and $h_n=-1$ and all other $h_i=0$, so that
$(\beta(4),\beta(6),\ldots,\beta(2k+2))$ must be a permutation of $\{1,2,\ldots,k\}$ and $(\beta(5),\beta(7),\ldots,\beta(2j+3))$ must be a permutation of $\{k+3,k+4,\ldots,k+j+2\}$ and $\beta(3)=k+1$ and $\beta(2)=k+2$ and $\beta(1)=k+j+3,$
%
\item $h_1=h_2=h_4\cdots=h_{2k+2}=-2$ and all other $h_i=-1$, so that
$(\beta(4),\beta(6),\break \ldots,\beta(2k+2))$ must be a permutation of $\{n-1,n-2,\ldots,n-k\}$  and $(\beta(5),\beta(7),\ldots,\beta(2j+3))$ must be a permutation of $\{n-k-3,n-k-4,\ldots,n-k-j-2\}$ and $\beta(3)=n-k-1$ and $\beta(2)=n-k-2$ and $\beta(1)=n-k-j-3.$
\end{itemize}
In each case $\beta$ can be extended in $(n-(k+j+4))!$ ways, giving the possible relative positions of the $(n-(k+j+4))$ $x_i$ (see Figure \ref{fig:configurationS2.C4 .S2}, for the case of $S_2\cdot C_4\cdot S_2$). So, there are $2\cdot k!\,j!\,(n-(k+j+4))!$ RH-configurations $(h, \beta).$
%%%%%%%%%%%
\begin{figure}[H]
\begin{center}
\scalebox{.54}{\input{configurationS2.C4.S2.pstex_t}}
\caption{Fractional representation of a simplicial subpolytope of $\Po_{RH}(K_n\backslash( S_2\cdot C_4\cdot S_2))$}
\label{fig:configurationS2.C4 .S2}
\end{center}
\end{figure}
%%%%%%%%%
\end{proof}
%%%%%%%%% K_n(K_3.S_k)
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{The Ree--Hoover weight of the graph $K_n\backslash (S_k \cdot K_3)$}
Let $S_k \cdot K_3$ denote the graph obtained by identifying one vertex of the graph $K_3$ with the centre of a $k$-star. See  Figure \ref{S4K3} for an example. 
$w_{RH}(K_n\backslash (S_k \cdot K_3))=0,$ for $k \geq 0,$ $n \geq k+5,$ which is a consequence of Theorem~\ref{theo:wRH=0}. Note that the special case $k=0$ corresponding to the graph $K_n\backslash K_3$.
% %% figure 
\begin{figure}[ht]
\vspace{.2cm}
\centerline{\includegraphics[width=.21\textwidth]{S4.K3ne.eps}}
 \caption{The graph $S_4\cdot K_3$}
 \label{S4K3}
 \end{figure}
% %
 %%%%%%%%% K_n(C_p.S_k)
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{The Ree--Hoover weight of the graph $K_n\backslash (S_k \cdot C_p)$}
Let $S_k \cdot C_p$ denote the graph obtained by identifying one vertex of the graph $C_p$ with the centre of a $k$-star. See  Figure \ref{S4C5} for an example. 
$w_{RH}(K_n\backslash (S_k \cdot C_p))=0,$ for $p \geq 5,$ $k \geq 1,$ $n \geq k+p,$ which is a consequence of Theorem~\ref{theo:wRH=0}. 
 %% figure 
\begin{figure}[ht]
\vspace{.2cm}
\centerline{\includegraphics[width=.31\textwidth]{C8.S4ne.eps}}
 \caption{The graph $S_4\cdot C_5$}
 \label{S4C5}
 \end{figure}
 %
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Ree--Hoover weight of the graph $K_n\backslash kS_1$}
%%%%%%%%%%%%%%    K-n\ke                  %%%%%%%%%%%

  For $k\geq2$,  $n\geq 2k$, let $K_n\backslash kS_1$ denote the complete graph on $n$ vertices from which $k$ separate edges have been removed, with $S_1=e$ is the graph with only one edge. 
  See  Figure \ref{3e} for an example. 
  Then we have $w_{RH}(K_n\backslash kS_1)=w_{RH}(K_n\backslash ke)= 0,$ which is a consequence of Theorem~\ref{theo:wRH=0}. 
%
%%% figure 
%\begin{figure}[ht]
%\vspace{.2cm}
%\centerline{\includegraphics[width=.16\textwidth]{3S1.eps}}
% \caption{The graph $3S_1$}
% \label{3e}
% \end{figure}
 %
% figure 
\begin{figure}[ht]
\vspace{.2cm}
\centerline{\includegraphics[width=.12\textwidth]{3S1ne.eps}}
 \caption{The graph $3S_1$}
 \label{3e}
 \end{figure}
 %
 
\medskip
 Note that, for $k=1,$ we have the following result.
 %% prop
\begin{proposition}[\cite{KP2}]
\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)|=|w_{RH}(K_n\backslash S_1)|= {\frac 2{(n-1)}}\cdot
\end{equation}
\end{proposition}

%
%proof prop 
\begin{proof}  See \cite{KP2}.
\end{proof}
Note that the formula (\ref{eq:Knmoinse}) is a special case of (\ref{eq:wRH Sk}).
%%%%%%%%%%%%%%%%        S_j K_3 S_k      %%%%%%%%%%%%%%%%%%%
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{The Ree--Hoover weight of the graph $K_n\backslash (S_j\cdot K_3\cdot S_k)$}
 Let  $(S_j\cdot K_3\cdot S_k)$ denote the graph obtained by joining with an edge of the graph $K_3$ the centres of a $j$-star and of a $k$-star. See  Figure \ref{S4K3S3} for an example. $w_{RH}(K_n\backslash (S_j\cdot K_3\cdot S_k))=0,$ for $j \geq k \geq 1,$ $ n \geq k+j+4,$ which is a consequence of Theorem~\ref{theo:wRH=0}.
 
\medskip
  %% figure 
\begin{figure}[ht]
\vspace{.2cm}
\centerline{\includegraphics[width=.31\textwidth]{S4.K3.S2ne.eps}}
 \caption{The graph $S_3\cdot K_3\cdot S_4$}
 \label{S4K3S3}
 \end{figure}
 %
%%%%%%%%%%%%%%%%        S_j C_p S_k      %%%%%%%%%%%%%%%%%%%
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{The Ree--Hoover weight of the graph $K_n\backslash (S_j\cdot C_p\cdot S_k)$}
 Let  $(S_j\cdot C_p\cdot S_k)$ denote the graph obtained by joining with an edge of the graph $C_p$ the centres of a $j$-star and of a $k$-star. See  Figure \ref{S2C6S4} for an example. $w_{RH}(K_n\backslash (S_j\cdot C_p\cdot S_k))=0,$ for $p \geq 5,$ $j \geq k \geq 1,$ $ n \geq k+j+p+1,$ which is a consequence of Theorem~\ref{theo:wRH=0}.
 
\medskip
  %% figure 
\begin{figure}[ht]
\vspace{.2cm}
\centerline{\includegraphics[width=.31\textwidth]{S3.C5.S4ne.eps}}
 \caption{The graph $S_2\cdot C_6\cdot S_4$}
 \label{S2C6S4}
 \end{figure}
 % \\
 
\medskip
We need to use Propositions~\ref{prop:Sk}--\ref{prop:C4.Sk} to prove Mayer's weight formulas that will be presented in Section~\ref{sec: newMayer}.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \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 $\{\{1, 2\}, \{1, 3\},\ldots,
\{1, k+1\}\}$, (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}  
%
 
%%%%%%%%%%%%%%    S_k  %%%%%%%%%%%%%%%
\begin{proposition}[\cite{KP2}]
\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 )}.
\end{equation}
\end{proposition}

% proof
%\begin{proof} See \cite{KP2}.
%\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 centres 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}
  %%%%%%%%%%%%%%%%           Sj-SK        %%%%%%%%%%%%%%
\begin{proposition}[\cite{KP2}]
\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))}.
 \end{equation}
\end{proposition}
% preuve 
%\begin{proof}See \cite{KP2}.
%\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 centre 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
 %%%%%%%%%%%%%%%        C_4 -Sk           %%%%%%%%%%%%%%%%%%%%%%%%
\begin{proposition}[\cite{KP2}]
\label{prop:C4.Sk}
For $k \geq 1,$ $ n \geq k+5,$ we have
%\vspace{-1.4045cm}
\begin{equation}\label{eq:wRH C4.Sk}
 | 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} See \cite{KP2}.
%\end{proof}
Note that the formula (\ref{eq:wRH C4.Sk}) is not a special case of (\ref{eq:wRH Sj.C4.Sk}).

%We need propositions (\ref{prop:Sk})-(\ref{prop:C4.Sk}) to prove Mayer's weight formulas in section \ref{sec: newMayer}.
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Section 3 :Mayer 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 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 gas 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.
%%%%%%%%%%%%%        C_k            %%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Mayer weight of the graph $K_n\backslash C_k$}
%
\begin{proposition}[\cite{KP2}]
%\label{propC4}
For $k=4$, $n \geq 6,$ we have
%
\begin{equation}
\label{eq:wM C4} | 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}
%See (\cite{KP2}).
%\end{proof}
%%%%%%
\begin{proposition}
%\label{propC4}\newpage
For $n \geq k \geq 5$ we have
%
\begin{equation}
\label{eq:wM Ck} | w_M(K_n\backslash C_k)|=n+\frac{2k}{n-1}
+\frac{4k}{(n-1)(n-2)}+\frac{2k}{(n-1)(n-2)(n-3)}.
\end{equation}
%
\end{proposition}

% proof
\begin{proof}
The supergraphs of $K_n\backslash C_k,$ $k \geq 5,$ whose Ree--Hoover weight is not zero are up to isomorphism of the form $K_n\backslash S_1$, $K_n\backslash S_2,$  $K_n\backslash(S_1$--$S_1)$ and $K_n.$ 
Their multiplicities are given by the formula 
%
\begin{multline*}
 |w_{M}(K_n\backslash C_k)|= |w_{RH}(K_n)|+\sum_{l=1}^{2}k |w_{RH}(K_n\backslash S_l)|
  +k|w_{RH}(K_n\backslash (S_1-S_1))|.
%\label{eq:proof C4}
\end{multline*}
%\newpage
We conclude using Propositions~\ref{prop:Sk} and~\ref{propSj-Sk}.
%
\end{proof}
Note that the formula (\ref{eq:wM C4}) is not a special case of (\ref{eq:wM Ck}).
%%%%%%%%%%%%%%%        Sj-C_4 -Sk           %%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Mayer weight of the graph $K_n\backslash (S_j \cdot C_4\cdot S_k)$}
%
\begin{proposition}
For $j \geq k \geq 1,$ $ n \geq k+j+5,$ we have, with the usual convention ${\binom {k+1} \ell}=0$ if $\ell>k+1$,
%
{\begin{align*}
 \notag 
| w_{M}(K_n\backslash
(S_j \cdot C_4\cdot S_k))| & = {{n+\sum_{l=1}^{j+2}2\left[\binom{j+2}{l}+\binom{k+2}{l}\right]\frac{l!}{\left(n-1\right)\cdots\left(n-l\right)}}}\\
 &   +\frac{8}{\left(n-1\right)\left(n-2\right)}+\frac{10}{\left(n-1\right)\left(n-2\right)\left(n-3\right)}\\
 &   +\sum_{l=1}^{j}4\left[\binom{j}{l}+\binom{k}{l}\right]\frac{l!}{\left(n-1\right)\cdots\left(n-l-3\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-3\right)}\\
 &   +\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-2\right)}\\
 &   +\sum_{m=1}^{j+1}\sum_{l=1}^{k+1}2\binom{j+1}{m}\binom{k+1}{l}\frac{m!\,l!}{\left(n-1\right)\cdots\left(n-m-l-1\right)} .
 \end{align*}}
%
\end{proposition}
% proof
\begin{proof}
The supergraphs of $K_n\backslash (S_j \cdot C_4\cdot S_k)$ whose Ree--Hoover weight is not zero are up to isomorphism of the form $K_n\backslash S_l$, $1\leq l \leq j+2,$ 
$K_n\backslash (C_4\cdot S_l),$ $1\leq l \leq j,$ $K_n\backslash(S_m$--$S_l),$ $1\leq m \leq j+1,$ $1\leq l \leq k+1,$ $K_n\backslash (S_m \cdot C_4\cdot S_l)$ $1\leq m \leq j,$ $1\leq l \leq k,$ $C_4$ and $K_n.$ Their multiplicities are given by the formula
%
%\newpage
{\begin{align*}
 \notag 
| w_{M}(K_n\backslash
(S_j \cdot C_4\cdot S_k))| & = |w_{RH}(K_n)|+\sum_{l=1}^{j+2}\left[\binom{j+2}{l}+\binom{k+2}{l}\right]|w_{RH}(K_n\backslash S_l)|\\
 & +2|w_{RH}(K_n\backslash S_2)|+|w_{RH}(K_n\backslash C_4)|+ |w_{RH}(K_n\backslash S_1-S_1)|\\
 & +\sum_{l=1}^{j}\left[\binom{j}{l}+\binom{k}{l}\right]|w_{RH}(K_n\backslash C_4 \cdot S_l)|\\
 & +\sum_{m=1}^{j}\sum_{l=1}^{k}\binom{j}{m}\binom{k}{l}|w_{RH}(K_n\backslash S_m\cdot C_4 \cdot S_l)|\\
 & +\sum_{l=1}^{j+1}\left[\binom{j+1}{l}+\binom{k+1}{l}\right]|w_{RH}(K_n\backslash (S_1-S_l))|\\
 & +\sum_{m=1}^{j+1}\sum_{l=1}^{k+1}\binom{j+1}{m}\binom{k+1}{l}|w_{RH}(K_n\backslash (S_m-S_l))|. 
 \end{align*}}
We conclude using Propositions~\ref{propC4} and \ref{prop:Sj.C4.Sk}--\ref{prop:C4.Sk}.
%
\end{proof}
%%%%%%%%%%%%%%%%           SK.K3        %%%%%%%%%%%%%%
\subsection{The Mayer weight of the graph $K_n\backslash (S_k\cdot K_3)$}
\begin{proposition}
%\label{propSj-Sk}
For $k \geq 0,$ $n \geq k+5,$ we have
%let $g_n=K_n\backslash (S_j$--$ S_k).$ Then we have
%
\begin{align*}%\label{eq:wM SkK3}
 | w_M(K_n\backslash (S_k\cdot K_3)|& =  n+\sum_{l=1}^{k+2}2\binom{k+2}{l} \frac{l!}{\left(n-1\right)\cdots\left(n-l\right)}\\
 &\kern2cm +  \sum_{l=1}^{k}4\binom{k}{l} \frac{l!}{\left(n-1\right)\cdots\left(n-l-2\right)} \\
 &\kern2cm +  \frac{2}{\left(n-1\right)}+\frac{8}{(n-1)(n-2)}.
 \end{align*}
 %Their multiplicities are given by
\end{proposition}

% proof
\begin{proof}
The supergraphs of $K_n\backslash (S_k\cdot K_3)$ whose Ree--Hoover weight is not zero are up to isomorphism of the form $K_n\backslash S_l$, $1\leq l \leq k+2,$ $K_n\backslash(S_1$--$S_l),$ $1\leq l \leq k$ and $K_n.$ Their multiplicities are given by the formula 
%
\begin{align*}
\notag |w_{M}(K_n\backslash (S_k\cdot K_3))|&= |w_{RH}(K_n)|+\sum_{l=1}^{k+2}\binom{k+2}{l}|w_{RH}(K_n\backslash S_l)|\\
 &\kern2cm +  \sum_{l=1}^{k}2\binom{k}{l}|w_{RH}(K_n\backslash (S_1-S_l))| \\
 &\kern2cm +  |w_{RH}(K_n\backslash S_1)|+2|w_{RH}(K_n\backslash S_2)|.
%\label{eq:proof Sk-Sj}
\end{align*}
We conclude using Propositions~\ref{prop:Sk} and \ref{propSj-Sk}.
%
\end{proof}
For the special case $k=0$, $n \geq 5$, we have
\begin {equation}
|w_{M}(K_n\backslash K_3)|= n+\frac{6}{\left(n-1\right)}+\frac{12}{(n-1)(n-2)}.
\end {equation}
%
%%%%%%%%%%%%%%%%           SK.Cp        %%%%%%%%%%%%%%
\subsection{The Mayer weight of the graph $K_n\backslash (S_k\cdot C_p)$}
\begin{proposition}
%\label{propSj-Sk}
For $p \geq 5,$ $k \geq 1,$ $n \geq k+p,$ we have
%let $g_n=K_n\backslash (S_j$--$ S_k).$ Then we have
%
\begin{align*}%\label{eq:wM SkCp}
| w_M(K_n\backslash (S_k\cdot C_p)|& =  n+\sum_{l=1}^{k+2}2\binom{k+2}{l} \frac{l!}{\left(n-1\right)\cdots\left(n-l\right)}\\
     &\kern1cm +  \sum_{l=1}^{k+1}4{\left[\binom{k}{l-1}+\binom{k}{l}\right] \frac{l!}{\left(n-1\right)\cdots\left(n-l-2\right)} }\\  
 &\kern1cm +  \frac{2(p-2)}{\left(n-1\right)}+\frac{4(p-1)}{(n-1)(n-2)}+\frac{2p}{(n-1)(n-2)(n-3)}.
 \end{align*}
 %
\end{proposition}

% proof
\begin{proof}
The supergraphs of $K_n\backslash (S_k\cdot C_p)$ whose Ree--Hoover weight is not zero are up to isomorphism of the form $K_n\backslash S_l$, $1\leq l \leq k+2,$ $K_n\backslash(S_1$--$S_l),$ $1\leq l \leq k+1$ and $K_n.$ Their multiplicities are given by the formula 
%
\begin{align*}
 |w_{M}(K_n\backslash (S_k\cdot C_p))|&= |w_{RH}(K_n)|+\sum_{l=1}^{k+2}\binom{k+2}{l}|w_{RH}(K_n\backslash S_l)|\\
 &\kern2cm +  \sum_{l=1}^{k+1}2\left[\binom{k}{l-1}+\binom{k}{l}\right]|w_{RH}(K_n\backslash (S_1-S_l))| \\
 &\kern2cm +  (p-2)|w_{RH}(K_n\backslash S_1)|+(p-1)|w_{RH}(K_n\backslash S_2)|\\
 &\kern2cm +  p|w_{RH}(K_n\backslash (S_1-S_1))|.
%\label{eq:proof SkCpj}
\end{align*}
We conclude using Propositions~\ref{prop:Sk} and \ref{propSj-Sk}.
%?\newpage
\end{proof}
%
%%%%%%%%%%%%%%%        Sj-K_3 -Sk           %%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Mayer weight of the graph $K_n\backslash (S_j \cdot K_3\cdot S_k)$}
%
\begin{proposition}
For $j \geq k \geq 1,$ $ n \geq k+j+4,$ we have, with the usual convention ${\binom k \ell}=0$ if $\ell>k$,
%\newpage
{\begin{align*}
 | w_{M}(K_n\backslash
(S_j \cdot K_3&{}\cdot S_k))|  =  {{n+\sum_{l=1}^{j+2}2\left[\binom{j+2}{l}+\binom{k+2}{l}\right]\frac{l!}{\left(n-1\right)\cdots\left(n-l\right)}}}\\
 & {{{}-\frac{2}{\left(n-1\right)}+\frac{4}{\left(n-1\right)\left(n-2\right)}}}\\
 &   {{+\sum_{l=1}^{j}4\left[\binom{j}{l}+\binom{k}{l}\right]\frac{l!}{\left(n-1\right)\cdots\left(n-l-2\right)}}}\\
 &   {{+\sum_{m=1}^{j+1}\sum_{l=1}^{j}2\left[\binom{j}{m}\binom{k}{l}+\binom{j}{m-1}\binom{k}{l}+\binom{j}{l}\binom{k}{m-1}\right]}}\\
 &\kern2cm
\cdot   \frac{m!\,l!}{\left(n-1\right)\cdots\left(n-m-l-1\right)}.
  \end{align*}}
%
\end{proposition}
% proof
\begin{proof}
The supergraphs of $K_n\backslash (S_j \cdot K_3\cdot S_k)$ whose Ree--Hoover weight is not zero are up to isomorphism of the form $K_n\backslash S_l$, $1\leq l \leq j+2,$ 
$K_n\backslash(S_m$--$S_l),$ $1\leq m \leq j+1,$ $1\leq l \leq k+1$ and $K_n.$ Their multiplicities are given by the formula 
%
%\newpage
{\begin{align*}
  | w_{M}(K_n\backslash
(S_j \cdot K_3\cdot S_k&))|  =  |w_{RH}(K_n)|+\sum_{l=1}^{j+2}\left[\binom{j+2}{l}+\binom{k+2}{l}\right]|w_{RH}(K_n\backslash S_l)|\\
 &   -|w_{RH}(K_n\backslash S_1)|+|w_{RH}(K_n\backslash S_2)|\\
 &   +\sum_{l=1}^{j}2\left[\binom{j}{l}+\binom{k}{l}\right]|w_{RH}(K_n\backslash (S_1-S_l))|\\
 &   +\sum_{m=1}^{j+1}\sum_{l=1}^{j}\left[\binom{j}{m}\binom{k}{l}+\binom{j}{m-1}\binom{k}{l}+\binom{j}{l}\binom{k}{m-1}\right]\\
 &\kern3cm
\cdot   |w_{RH}(K_n\backslash (S_m-S_l))|. 
 \end{align*}}
We conclude using Propositions~\ref{prop:Sk} and \ref{propSj-Sk}.
%
\end{proof}
%%%%%%%%%%%%%%%        Sj-C_p -Sk           %%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Mayer weight of the graph $K_n\backslash (S_j \cdot C_p\cdot S_k)$}
%
\begin{proposition}
For $p \geq 5,$ $j \geq k \geq 1,$ $ n \geq k+j+p+1,$ we have, with the usual convention ${\binom k \ell}=0$ if $\ell>k$,
%
{\begin{align*}
 | w_{M}(K_n\backslash
(S_j \cdot C_p\cdot S_k))| & = n+\sum_{l=1}^{j+2}2\left[\binom{j+2}{l}+\binom{k+2}{l}\right]\frac{l!}{\left(n-1\right)\cdots\left(n-l\right)}\\
 &  +\frac{2(p-4)}{\left(n-1\right)}+\frac{4(p-2)}{\left(n-1\right)\left(n-2\right)}+\frac{2(p-3)}{\left(n-1\right)\left(n-2\right)\left(n-3\right)}\\
 &  +\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-2\right)}\\
 & +\sum_{m=1}^{j+1}\sum_{l=1}^{k+1}2\binom{j+1}{m}\binom{k+1}{l}\frac{m!\,l!}{\left(n-1\right)\cdots\left(n-m-l-1\right)}.
  \end{align*}}
%
\end{proposition}
% proof
\begin{proof}
The supergraphs of $K_n\backslash (S_j \cdot C_p\cdot S_k)$ whose Ree--Hoover weight is not zero are up to isomorphism of the form $K_n\backslash S_l$, $1\leq l \leq j+2,$ 
$K_n\backslash(S_m$--$S_l),$ $1\leq m \leq j+1,$ $1\leq l \leq k+1$ and $K_n.$ Their multiplicities are given by the formula 
%
%\newpage
{\begin{align*}
 | w_{M}(K_n\backslash
(S_j \cdot C_p\cdot S_k))| & = |w_{RH}(K_n)|+\sum_{l=1}^{j+2}\left[\binom{j+2}{l}+\binom{k+2}{l}\right]|w_{RH}(K_n\backslash S_l)|\\
 &   +\sum_{l=1}^{j+1}\left[\binom{j+1}{l}+\binom{k+1}{l}\right]|w_{RH}(K_n\backslash (S_1-S_l))|\\
&  +\sum_{m=1}^{j+1}\sum_{l=1}^{k+1}2\binom{j+1}{m}\binom{k+1}{l}|w_{RH}(K_n\backslash (S_m-S_l))|\\
&  +(p-4)|w_{RH}(K_n\backslash S_1)|+(p-2)|w_{RH}(K_n\backslash S_2)|\\
 &  +(p-3)|w_{RH}(K_n\backslash (S_1-S_1))|.
 \end{align*}}
We conclude using Propositions~\ref{prop:Sk} and \ref{propSj-Sk}.
%
\end{proof}

%%%%%%%%%%%%%       kS_1         %%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Mayer weight of the graph $K_n\backslash kS_1$}
\begin{proposition}
\label{propkS1}
For $k \geq 1,$ $n \geq 2k,$ we have
%
\begin{equation*}
%\label{eq:wM kS1}
 | w_M(K_n\backslash kS_1)|=n+\frac{2k}{(n-1)}.
\end{equation*}
%
\end{proposition}

% proof
\begin{proof}
The supergraphs of $K_n\backslash kS_1,$ $k \geq 1,$  whose Ree--Hoover weight is not zero are up to isomorphism of the form $K_n\backslash S_1$ and $K_n.$ Their multiplicities are given by the formula 
%
%Hence,
\begin{equation*}
\notag |w_{M}(K_n\backslash kS_1)|= |w_{RH}(K_n)|+ k|w_{RH}(K_n\backslash S_1)|.
\end{equation*} 
We conclude using Proposition~\ref{prop:Sk}.
%
\end{proof}
%\vspace{-1cm}
%\newpage
%%%%%%%%%%%%%%%%%%%      bib            %%%%%%%%%%%%%%%%%%%%%
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\end{document}
