A Complete Solution to a Conjecture on the $\beta$-Polynomials of Graphs
Journal of Mathematical Chemistry, 33 (3):
189--193 (2003)DOI: 10.1023/A:1024738623798
Abstract
In 1990, Gutman and Mizoguchi conjectured that all roots of the $\beta$-polynomial $\beta$(G,C,x) of a graph G are real. Since then, there has been some literature intending to solve this conjecture. However, in all existing literature, only classes of graphs were found to show that the conjecture is true; for example, monocyclic graphs, bicyclic graphs, graphs such that no two circuits share a common edge, graphs without 3-matchings, etc, supporting the conjecture in some sense. Yet, no complete solution has been given. In this paper, we show that the conjecture is true for all graphs, and therefore completely solve this conjecture.