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On graphs whose flow polynomials have real roots only
Citation
Dong, F. (2018). On graphs whose flow polynomials have real roots only. Electronic Journal of Combinatorics, 25(3), Article P3.26. https://www.combinatorics.org/ojs/index.php/eljc/article/view/v25i3p26/pdf
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.
Date Issued
2018
Publisher
Electronic Journal of Combinatorics
Journal
Electronic Journal of Combinatorics