  In recent joint work (2021), we introduced a novel multivariate polynomial
attached to every metric space - in particular, to every finite simple
connected graph $G$ - and showed it has several attractive properties. First,
it is multi-affine and real-stable (leading to a hitherto unstudied
delta-matroid for each graph $G$). Second, the polynomial specializes to (a
transform of) the characteristic polynomial $\chi_{D_G}$ of the distance matrix
$D_G$; as well as recovers the entire graph, where $\chi_{D_G}$ cannot do so.
Third, the polynomial encodes the determinants of a family of graphs formed
from $G$, called the blowups of $G$.
  In this short note, we exhibit the applicability of these tools and
techniques to other graph-matrices and their characteristic polynomials. As a
particular case, we will see that the adjacency characteristic polynomial
$\chi_{A_G}$ is in fact the shadow of a richer multivariate blowup-polynomial,
which is similarly multi-affine and real-stable. Moreover, this polynomial
encodes not only the aforementioned three properties, but also yields
additional information for specific families of graphs.
