Mesure et action des i-permutations sur les multigraphes multicolores finis et inifinis

Among other results, the purpose of this article is to show the existence of an $\mathbb{R}$-space-vector with basis ${\omega }_j^i$, $i,j$ are integers such that every graph with n vertex $n\ge 3$ is the vector: $$\mathcal{V}(n)=\sum _{j=0}^{n-1}{\alpha }_j^{n-1}{\omega }_j^{n-1}$$ where ${\alpha }_j^{n-1}$ is the number of sub graphs of type ${\omega }_j^{n-1}$ . We deduce that two graphs are isomorphic if for any measure, they have the same number of maximal proper subset with this measure.