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Blowup Polynomials and delta-Matroids of Graphs

Choudhury, PN and Khare, A (2022) Blowup Polynomials and delta-Matroids of Graphs. In: Seminaire Lotharingien de Combinatoire (86).

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Official URL: https://doi.org/10.48550/arXiv.2203.04105


For every finite simple connected graph G = (V, E), we introduce an invariant, its blowup-polynomial pG({nv: v 2 V}). This is obtained by dividing the determinant of the distance matrix of its blowup graph G[n] (containing nv copies of v) by an exponential factor. We show that pG(n) is indeed a polynomial function in the sizes nv, which is moreover multi-affine and real-stable. This associates a hitherto unexplored delta-matroid to each graph G; and we provide a second novel one for each tree. We also obtain a new characterization of complete multipartite graphs, via the homogenization at -1 of pG being completely/strongly log-concave, i.e., Lorentzian. (These results extend to weighted graphs.) Finally, we show pG is indeed a graph invariant, i.e., pG and its symmetries (in the variables n) recover G and its isometries.

Item Type: Journal Article
Publication: Seminaire Lotharingien de Combinatoire
Publisher: Universitat Wien, Fakultat fur Mathematik
Additional Information: The copyright for this article belongs to the Universitat Wien, Fakultat fur Mathematik.
Keywords: blowup-polynomial; delta-matroid; distance matrix; real-stable polynomial; Zariski density
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 11 Jul 2023 06:31
Last Modified: 11 Jul 2023 06:31
URI: https://eprints.iisc.ac.in/id/eprint/82423

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