On Graphs whose Flow Polynomials have Real Roots Only

Fengming Dong

doi:10.37236/7512

Abstract

Let $G=(V,E)$ be a bridgeless graph. In 2011 Kung and Royle showed that the flow polynomial $F(G,\lambda)$ of $G$ has integral roots only if and only if $G$ is the dual of a chordal and plane graph. In this article, we study whether every graph whose flow polynomial has real roots only is the dual of some chordal and plane graph. We conclude that the answer for this problem is positive if and only if $F(G,\lambda)$ does not have any real root in the interval $(1,2)$. We also prove that for any non-separable and $3$-edge connected $G$, if $G-e$ is also non-separable for each edge $e$ in $G$ and every $3$-edge-cut of $G$ consists of edges incident with some vertex of $G$, then $P(G,\lambda)$ has real roots only if and only if either $G\in \{L,Z_3,K_4\}$ or $F(G,\lambda)$ contains at least $9$ real roots in the interval $(1,2)$, where $L$ is the graph with one vertex and one loop and $Z_3$ is the graph with two vertices and three parallel edges joining these two vertices.

Highlights

The graphs considered in this paper are undirected and finite, and may have loops and parallel edges
F (G, λ) = 0) whenever G contains loops, we will assume that G is loopless when P (G, λ)
Due to Tutte [20], P (G, λ) = λF (G∗, λ) holds for any connected plane graph G, where G∗ is the dual of G. It is unknown if there is a planar graph G which has real flow roots only and has non-integral flow roots

Summary

Introduction

The graphs considered in this paper are undirected and finite, and may have loops and parallel edges. Due to Tutte [20], P (G, λ) = λF (G∗, λ) holds for any connected plane graph G, where G∗ is the dual of G Equivalently, it is unknown if there is a planar graph G which has real flow roots only and has non-integral flow roots. By the following result due to Kung and Royle [13], Problem 1 is equivalent to whether there exists a graph G which is not the dual of any plane and chordal graph but has real flow roots only. Theorem 1 ([13]) If G is a bridgeless graph, its flow roots are integral if and only if G is the dual of a chordal and plane graph. (i) G is the dual of some chordal and plane graph; (ii) G does not have any flow root in the interval (1, 2); (iii) each flow root of G is in the set {1, 2, 3}

Basic results on flow polynomials

Findings

Graphs with real flow roots only