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 pX ({nx : x ∈X}). This is obtained from the blowup X[n] – which contains nx copies of each point x – by computing the determinant of the distance matrix of X[n] and removing an exponential factor. We prove that as a function of the sizes nx , pX (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 pG is “partially symmetric” in {nv : v ∈ V(G)} – and show three further results: (a) We show that the polynomial pG is indeed a graph invariant, in that pG and its symmetries recover the graph G and its isometries, respectively. (b) We show that the univariate specialization uG(x) := pG(x, . . ., x) is a transform of the characteristic polynomial of the distance matrix DG; 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 pG(n) is real-stable (equivalently, Lorentzian, or strongly/completely log-concave), if and only if the normalization of pG(−n) is strongly Rayleigh.