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 pX({ n(x) : x is an element of X }). This is obtained from the blowup X[n] - which contains n(x) 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 { n(v) : v is an element of 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 u(G)(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.