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

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. This is obtained from the blowup-which contains copies of each point x-by computing the determinant of the distance matrix of and removing an exponential factor. We prove that as a function of the sizes, 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 a connected unweighted graph-so is partially symmetric in-and show three further results: (a) We show that the polynomial is indeed a graph invariant, in that and its symmetries recover the graph G and its isometries, respectively. (b) We show that the univariate specialization is a transform of the characteristic polynomial of the distance matrix; 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 of is real-stable (equivalently, Lorentzian, or strongly/completely log-concave), if and only if the normalization of is strongly Rayleigh.