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