  For every finite simple connected graph $G = (V,E)$, we introduce an
invariant, its blowup-polynomial $p_G(\{ n_v : v \in V \})$. This is obtained
by dividing the determinant of the distance matrix of its blowup graph $G[{\bf
n}]$ (containing $n_v$ copies of $v$) by an exponential factor. We show that
$p_G({\bf n})$ is indeed a polynomial function in the sizes $n_v$, which is
moreover multi-affine and real-stable. This associates a hitherto unexplored
delta-matroid to each graph $G$; and we provide a second unexplored one for
each tree. As another consequence, we obtain a new characterization of complete
multipartite graphs, via the homogenization at $-1$ of $p_G$ being
completely/strongly log-concave, i.e., Lorentzian. (These results extend to
weighted graphs.) Finally, we show $p_G$ is indeed a graph invariant, i.e.,
$p_G$ and its symmetries (in the variables ${\bf n}$) recover $G$ and its
isometries, respectively.
