The(weighted)harary Indices Of Graphs

All graphs considered in this thesis are undirected and simple. In chemical graph theory, topological index, also known as molecular descriptor, is used to describe the molecular graphs and some properties of invariant. Harary index of graphs, which is defined on the basis of distance, is a topological index. It can be denoted as H(.It has been introduced independently by et al. and et al. in 1993.In this thesis we mainly study the mathematical properties of Harary index of graphs and its modified weighted indices(such as additivity weighted Harary index, multiplicatively weighted Harary index). We obtain the upper and lower bounds on these indices of some special graphs and characterize the extremal graphs at which the upper and lower bounds of the(weighted) Harary indices. In the first chapter we briefly recall the basic concepts in graph theory and the related research background. In the second chapter some upper and lower bounds on Harary index are determined for non-trivial quasi-tree graphs and generalized quasi-tree graphs. In the third chapter, we consider the additively weighted Harary index and obtain the upper and lower bounds on the additively weighted Harary index for non-trivial quasi-tree graphs、generalized quasi-tree and some(n,m)-graphs. Moreover, we characterize the corresponding extremal graphs which reach the upper and lower bounds on these indices. In the fourth chapter, we discuss the properties of multiplicative Harary index and determine the upper and lower bounds on the multiplicatively weighted Harary index in non-trivial quasi-tree graphs, generalized quasi-tree graphs and some(n,m)-graphs, respectively. Furthermore the corresponding extermal graphs are characterized which attain the upper and lower bounds on these indices. In the fifth chapter the results obtained in this thesis are summarized and some related open problems are listed.