Some results on an algebro-geometric condition on graphs

TL;DR

This paper investigates an algebro-geometric condition related to Kirchhoff polynomials of graphs, providing characterizations and results for specific graph classes and edge configurations.

Contribution

It offers new insights into when Kirchhoff polynomials of graphs satisfy a particular Jacobian ideal condition, including characterizations for wheel and series-parallel graphs.

Findings

Multiple edges can be reduced to double edges.

Characterization of edges in wheel graphs satisfying the property.

Identification of a class of series-parallel graphs with the property.

Abstract

Paolo Aluffi, inspired by an algebro-geometric problem, asked when the Kirchhoff polynomial of a graph is in the Jacobian ideal of the Kirchhoff polynomial of the same graph with one edge deleted. We give some results on which graph-edge pairs have this property. In particular we show that multiple edges can be reduced to double edges, we characterize which edges of wheel graphs satisfy the property, we consider a stronger condition which guarantees the property for any parallel join, and we find a class of series-parallel graphs with the property.