Research On Resistance-Harary Index Among Unicyclic And Bicyclic Graphs

In 1993,Klein and Randic introduced the distance function named resistance distance based on the theory of electrical network.They viewed a graph G as an electrical network N,where each edge of G is assumed to be a unit resistor.The resistance distance between vertices u and v in G,denoted by rG(u,v),is defined to be the equivalent resistance between nodes u and v of N.The resistance-Harary index RH(G)of a connected graph is a graph invariant based on resistance distance,defined as the sum of reciprocal resistance distances between all vertex pairs.In this thesis,we mainly studied the resistance-Harary index of unicyclic graphs and oo-type bicyclic graphs,and characterized the corresponding extremal graphs.The main work includes the following chapters:In the first chapter,we mainly introduced the background and significance of our research,as well as the relevant representative research results at home and abroad,and also briefly introduced the main research contents of this thesis.In the second chapter,we gave some concepts,terms,notations,and related lemmas.In the third chapter,we determined the range of the possible extreme graphs with mimimum resistance Harary index among all the unicyclic graphs and oo-type bicyclic graphs of n vertices through graph operations.Then,we characterized the minimum re-sistance Harary index among all the n-vertex unicyclic graphs and ?-type bicyclic graph-s,respectively,by recursion and verification.Furthermore,we also obtained the extreme graphs with the second minimum resistance Harary index among all the n-vertex unicyclic graphs and oo-type bicyclic graphs,respectively.In the fourth chapter,we summarized the main results of this thesis,and put forward some further research directions on this basis.