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 ow polynomial F(G; ) 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; ) 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; ) 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; ) are real if and only if either G 2 fL;Z3;K4g or F(G; ) contains at least 9 real roots in the interval (1; 2), where L is the graph with one vertex and one loop and Z3 is the graph with two vertices and three parallel edges joining these two vertices.

Mathematics Subject Classi cations: 05C21, 05C31