Choudhury, Projesh NathProjesh NathChoudhuryKhare, ApoorvaApoorvaKhare2025-08-312025-08-312024-12-0110.4153/S0008414X230007312-s2.0-85176454740https://d8.irins.org/handle/IITG2025/26419To 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.trueblowup-polynomial | delta-matroid | distance spectrum of a Graph | real-stable polynomial | Zariski densityArticlehttps://www.cambridge.org/core/services/aop-cambridge-core/content/view/F6D9B3E881EE2AE9BE53A8E3B5DBBA98/S0008414X23000731a.pdf/div-class-title-the-blowup-polynomial-of-a-metric-space-connections-to-stable-polynomials-graphs-and-their-distance-spectra-div.pdf2073-21141 December 20240arJournal0WOS:001137018200001