Graphs whose flow polynomials have only integral roots

TL;DR

This paper characterizes graphs with integral flow polynomial roots, showing they are dual to planar chordal graphs, and extends the result to certain cubic graphs with real roots, linking matroid theory and graph planarity.

Contribution

It establishes a novel characterization of graphs with integral flow roots as duals of planar chordal graphs, connecting graph theory and matroid properties.

Findings

Graphs with integral flow roots are dual to planar chordal graphs.

For 3-connected cubic graphs, real flow roots imply the same duality.

Cographic matroids with integral characteristic roots correspond to cycle matroids of planar chordal graphs.

Abstract

We show if the flow polynomial of a bridgeless graph G has only integral roots, then G is the dual graph to a planar chordal graph. We also show that for 3-connected cubic graphs, the same conclusion holds under the weaker hypothesis that it has only real flow roots. Expressed in the language of matroid theory, this result says that the cographic matroids with only integral characteristic roots are the cycle matroids of planar chordal graphs.