  Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis
generating functions for all matroids. From the viewpoint of combinatorial
Hodge theory, it is natural to ask whether these functions are "strictly"
log-concave for simple matroids. In this paper, we show this strictness for
simple graphic matroids, that is, we show that Kirchhoff polynomials of simple
graphs are strictly log-concave. Our key observation is that the Kirchhoff
polynomial {of a complete graph} can be seen as the (irreducible) relative
invariant of a certain prehomogeneous vector space, which may be independently
interesting in its own right. Furthermore, we prove that for any
$a_i\in\mathbb{R}_{>0}$, $a_1x_1+\cdots+a_nx_n\in R^1_{M}$ satisfies the strong
Lefschetz property (moreover, Hodge--Riemann bilinear relation) at degree one
of the Artinian Gorenstein algebra $R^*_M$ associated to a graphic matroid $M$,
which is defined by Maeno and Numata for all matroids.
