The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra

Abstract

To every finite metric space $X$, including all connected unweighted graphs with the minimum edge-distance metric, we attach an invariant that we call its blowup-polynomial $p_X(\{ n_x : x \in X \})$. This is obtained from the blowup $X[{\bf n}]$ - which contains $n_x$ copies of each point $x$ - by computing the determinant of the distance matrix of $X[{\bf n}]$ and removing an exponential factor. We prove that as a function of the sizes $n_x$, $p_X({\bf n})$ is a polynomial, is multi-affine, and is real-stable. This naturally associates a hitherto unstudied delta-matroid to each metric space $X$; we produce another novel delta-matroid for each tree, which interestingly does not generalize to all graphs. We next specialize to the case of $X = G$ a connected unweighted graph - so $p_G$ is "partially symmetric" in $\{ n_v : v \in V(G) \}$ - and show three further results: (a) We show that the polynomial $p_G$ is indeed a graph invariant, in that $p_G$ and its symmetries recover the graph $G$ and its isometries, respectively. (b) We show that the univariate specialization $u_G(x) := p_G(x,\dots,x)$ is a transform of the characteristic polynomial of the distance matrix $D_G$; this connects the blowup-polynomial of $G$ to the well-studied "distance spectrum" of $G$. (c) We obtain a novel characterization of complete multipartite graphs, as precisely those for which the "homogenization at $-1$" of $p_G({\bf n})$ is real-stable (equivalently, Lorentzian, or strongly/completely log-concave), if and only if the normalization of $p_G(-{\bf n})$ is strongly Rayleigh.