The Harary Index Extreme Value Problem Of The Circle Graph

The research of topological indexs of molecular graph is an important topic in chemical graph theory,and its results are mainly applied to the research of the properties of molecular structure in the chemical.In this thesis,we discussed one of the topological index with great concern to scholars--Harary index,obtained the extremum and extremal graph of the Harary index of a circle graph while given the number of cycles.In regard to the maximum value,some scholars have researched on the maximal value of the Harary index of the general connected graph and the circle graph in the references,and the influence of the number of sides on the Harary index of the connected graph.Based on these references,we described the extremal graph structure corresponding to the maximal value of the Harary index of the multi graph.Considering the minimum value,based on the decrease of the Harary index of cyclic graphs,we put forward five effective graphics transformations,and quickly push out the minimal graphs of Harary index of the circle graph with no more than three circle with the five transformation.And we calculate the minimum value of the Harary index according to the tiny graph structure.At the end of the paper,a constitution of the tiny graph of the Harary index minigraph of the multi graph and the conjecture of research ideas are given.In the first chapter,we introduced the research background of the Harary index,gave some basic knowledge and relevant status,and explained the main conclusions of this paper.The second chapter is the detailed discussion and proof of the conclusion of this paper.On the one hand,We combine the lemma to give the upper bound of the Harary index of the k-circle graph,and get the maximum graph of the Harary index of k-circle graphs.On the other hand,we put forward five effective graphics transformations based on the decrease of the Harary index of cyclic graphs and use above-mentioned transformation quickly to push out the minimal graphs of Harary index of the circle graph with no more than three circle,and the Harary index is calculated according to the structure of the tiny graph.