  Let us consider a truncated matroid $M_{\Gamma}^{r}$ of rank $r$ of a graphic
matroid of a graph $\Gamma$. The basis for $M_{\Gamma}^{r}$ is the set of the
forests with $r$ edges in $\Gamma$. We consider this basis generating function
and compute its Hessian. In this paper, we show that the Hessian of the basis
generating function of the truncated matroid of the graphic matroid of the
complete or complete bipartite graph does not vanish by calculating the
eigenvalues of the Hessian matrix. Moreover, we show that the Hessian matrix of
the basis generating function of the truncated matroid of the graphic matroid
of the complete or complete bipartite graph has exactly one positive
eigenvalue. As an application, we show the strong Lefschetz property for the
Artinian Gorenstein algebra associated to the truncated matroid.
