A combinatorial matrix approach for the generation of vacuum Feynman graphs multiplicities in $\phi^{4}$ theory

From the standard procedure for constructing Feynman vacuum graphs in phi(4) theory from the generating functional Z, we find a relation with sets of certain combinatorial matrices, which allows us to generate the set of all Feynman graphs and the respective multiplicities in an equivalent combinatoric way. These combinatorial matrices are explicitly related with the permutation group, which facilitates the construction of the vacuum Feynman graphs. Various insights in this combinatoric problem are proposed, which in principle provide an efficient way to compute Feynman vacuum graphs and their multiplicities.