Research On Some Problems Of Topological Index And Structural Property Of Graphs

A chemical structure of a molecule can be easily represented by a graph which is named as chemical graph or molecular graph. A molecular topological index is a mapping from a set of graphs to a set of real numbers,it's a very significant parameter to characterize and measure the topological properties of chemical molecules.This topological index not only can be used to determine the structure of a molecule,but also can build the relationship between molecular structure and property.This paper mainly focuses on the relationship between a graph's structure and Forgotten index.Forgotten index is a topological index raised by I.Gutman in 1972,it calculates the cubic sum of a graph's vertex degree.In the first chapter,we introduce some basic concepts and some common marks in graph theory.We look back to those classic problems during the development history of graph theory. And also we introduce the related knowledge of topological indices.In the second chapter,we give the definition of clique and clique number of a graph and we establish the upper and lower bound of Forgotten index for a set of graphs with fixed clique number.In the third chapter,we give the definition of connectivity of a graph and we establish the upper and lower bound of Forgotten index for a set of graphs with fixed vertex connectivity or edge connectivity.In the forth chapter,we give the definition of matching number of a graph and we establish the upper and lower bound of Forgotten index for a set of graphs with fixed matching number.In the fifth chapter,we mainly summarize this paper and we list some unsolved problems.