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

Choudhury, PN and Khare, A (2023) The blowup-polynomial of a metric space: connections to stable polynomials, graphs and their distance spectra. In: Canadian Journal of Mathematics .

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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 2 X g. 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 fnv : v 2 V G 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 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. © 2020 Canadian Mathematical Society.

Item Type:	Journal Article
Publication:	Canadian Journal of Mathematics
Publisher:	Cambridge University Press
Additional Information:	The copyright for this article belongs to author
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	04 Mar 2024 09:21
Last Modified:	04 Mar 2024 09:21
URI:	https://eprints.iisc.ac.in/id/eprint/84328

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