The Wiener Polarity Index Of Bicyclic Graphs

In this paper, we purse a point of view on the Wiener polarity index of theAlgebraic graphs systems. Specificly, we describe the Wiener polarity index ofbicyclic graphs.LetG (U A, t,UB)be the bicyclic graphs which contain two cyclesU AandU B,t denote the shortest length of common edges in the cyclesU AandU B.The Wienerpolarity index of G is defined byWP(G)={(u, v) dG(u,v)=3;u,v∈V}. The formulasabout the Wiener polarity index of all bicyclic graphs are classified completely,Moreover, we obtain a upper bound for the Wiener polarity index of bicyclic chemicalgraphs.In Chapter1, the author introduces the background and the development of thesubject chosen. Besides these, the main structure of this thesis are explained.In Chapter2, the author studies the Wiener polarity index of bicyclic graphs.The Wiener polarity index over trees and unicyclic graphs have been studied by anumber of authors and achieved some results. However, until now the study about theWiener polarity index over bicyclic graphs is almost blank. This is the mainmotivation of Chapter2. Using some important results and methods of Algebraicgraphs, the author characterizes the Wiener polarity index of bicyclic graphs and givesthe formulas about the Wiener polarity index of all bicyclic graphs.In Chapter3, the author studies the upper bound for the Wiener polarity index ofbicyclic graphs. Using the method of broken the cycles, the author put a bicyclicgraphs into two unicyclic graphs. And last, Using some known important results andrigorous calculation, the author characterizes the upper bound for the Wiener polarityindex of bicyclic chemical graphs.Core conclusions are summarized in the last chapter, accompanied with thedirection of the future research.