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arXiv · 2106.03751

Multivariate blowup-polynomials of graphs

Projesh Nath Choudhury·Apoorva Khare
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Abstract

In recent joint work (2021), we introduced a novel multivariate polynomial attached to every metric space - in particular, to every finite simple connected graph $G$ - and showed it has several attractive properties. First, it is multi-affine and real-stable (leading to a hitherto unstudied delta-matroid for each graph $G$). Second, the polynomial specializes to (a transform of) the characteristic polynomial $\chi_{D_G}$ of the distance matrix $D_G$; as well as recovers the entire graph, where $\chi_{D_G}$ cannot do so. Third, the polynomial encodes the determinants of a family of graphs formed from $G$, called the blowups of $G$. In this short note, we exhibit the applicability of these tools and techniques to other graph-matrices and their characteristic polynomials. As a particular case, we will see that the adjacency characteristic polynomial $\chi_{A_G}$ is in fact the shadow of a richer multivariate blowup-polynomial, which is similarly multi-affine and real-stable. Moreover, this polynomial encodes not only the aforementioned three properties, but also yields additional information for specific families of graphs.

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Projesh Nath Choudhury, Apoorva Khare. 2021-06-07. Multivariate blowup-polynomials of graphs. https://arxiv.org/abs/2106.03751

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