Shapley Distance And Shapley Index Of Graphs

Profit distribution is an important scientific issue in cooperative game.Shapley value,providing a fair profit distribution scheme,is of great significance to the cooperative game allocation problem.Based on Shapley value,the concept of Shapley distance on graphs is proposed.Shapley distance can be used to measure the cost of one vertex to access another one.This paper is divided into five chapters as follows.The first chapter is the preliminaries.Firstly,the basic concepts and notations of graph are introduced.Secondly,the definition of Shapley value and Shapley distance of graphs are introduced.Finally,similar to Wiener index and Kirchhoff index,a new graph parameter,namely Shapley index,is proposed.In the second chapter,we establish the expressions of Shapley distance and Shapley index of some basic graphs in graph theory,such as tree,cycle,complete bipartite graph and complete multipartite graph.In addition,the relationships among Shapley index,Wiener index and Kirchhoff index of these basic graphs are determined.In the third chapter,we consider the corona operation in the classical graph theory.The exact expressions of Shapley distance for any pair of nodes in some composite graphs,such as fan graph,wheel graph and sun graph,are derived.In addition,the upper and lower bounds of their Shapley index are suggested.In the fourth chapter,we establish the accurate expressions of Shapley distance and Shapley index of some concrete graphs through gluing operation,such as friendship graph,generalized rose graph and so on.In addition,the relationships among Shapley index,Wiener index and Kirchhoff index are presented.The fifth chapter summarizes the main research work of this paper and puts forward some problems worthy of further consideration.