On Graphs whose Flow Polynomials have Real Roots Only

Abstract

Let G be a bridgeless graph. In 2011 Kung and Royle showed that all roots of the flow polynomial F (G, λ) of G are integers if and only if G is the dual of a chordal and plane graph. In this article, we study whether a bridgeless graph G for which F (G, λ) has real roots only must be the dual of some chordal and plane graph. We conclude that the answer of this problem for G is positive if and only if F (G, λ) 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 all roots of P (G, λ) are real if and only if either G ∈ {L, Z 3 , K 4 } or F (G, λ) 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.Mathematics Subject Classifications: 05C21, 05C31 * This paper was partially supported by NTU AcRF project (RP 3/16 DFM) of Singapore.

A survey on the study of real zeros of flow polynomials

“…Problem asks if holds. Dong obtained the following results on the study of Problem .…”

“…By Lemmas and , to study Problem , it suffices to consider ‐edge connected non‐separable graphs in which do not contain any proper ‐edge‐cut. Dong also showed that if is such a graph in , then and contains at least flow roots in , where has values , respectively, for and for all integers .…”

“…So far it is unknown if there exists a bridgeless graph having real flow roots only which also containsnon-integral real flow roots. In [10], Dong studies the following problem.…”

Polynomials related to chromatic polynomials

“…(v) Theorem 4.6 (Dong [8]) For any bridgeless graph G, if F (G, x) has real roots only, then either all roots of F (G, x) are integral or F (G, x) has at least 9 roots in (1, 2).…”

“…(iv) Conjecture 4.4 (Dong [8,10]) For any bridgeless graph G, if F (G, q) has real roots only, then all roots of F (G, q) are integral.…”