Blowup polynomials and delta-matroids of graphs

Abstract

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.