Matching energy and hosoya index on the complements of unicycle graphs

Abstract The matching energy of a graph is defined as the sum total of the absolute values of all the zeros of its matching polynomial, which is a quantity of significance in chemical applications. Since the study of extreme matching energy is an interesting work, some authors have given some vastly amusing elementary results on this topic. For example, Gutman, Ji, Chen, etc. characterized the unicycle, bicycle, and tricycle graphs with extremal matching energy. In this article, we focus on the matching energy of the complements of graphs, and four relevant graph transformations are given, from which we completely determine the graphs with extremal matching energy, among all complements of trees and all complements of unicycle graphs, respectively. As a byproduct, we also find that there exist the same results for the extremal Hosoya index.