Multivariate blowup-polynomials of graphs

TL;DR

This paper explores a new multivariate polynomial associated with graphs that encodes various graph properties, including distance and adjacency matrices, and introduces the concept of blowups to analyze graph matrices more comprehensively.

Contribution

It introduces a multivariate blowup-polynomial for graphs that generalizes known polynomials and encodes extensive graph information, including new insights into adjacency matrices.

Findings

Polynomial is multi-affine and real-stable.

Encodes determinants of graph blowups.

Specializes to known graph polynomials and reveals new properties.

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.