Given a finite simple connected graph $G = (V,E)$, we introduce a novel invariant which we call its blowup-polynomial $p_G({n_v : v \in V})$. To do so, we compute the determinant of the distance matrix of the graph blowup, obtained by taking $n_v$ copies of the vertex $v$, and remove an exponential factor. In this talk, we first see that as a function of the sizes $n_v$, $p_G$ is a polynomial, is multi-affine, and is real-stable. Second: we show that the multivariate polynomial $p_G$ is intimately related to the characteristic polynomial $q_G$ of the distance matrix $D_G$, and that it fully recovers $G$ whereas $q_G$ does not. Third: we obtain a novel characterization of the complete multi-partite graphs, as precisely those whose “homogenized” blowup-polynomials are Lorentzian/strongly Rayleigh. Finally: we explain how to obtain from $p_G$ a novel delta-matroid for every graph; we also provide a second delta-matroid for every tree, which too is hitherto unexplored, but whose construction does not extend to all graphs. (Joint with Apoorva Khare.)