The Research On Two Classes Of Topological Indices Of Graphs

Searching the moleculars with certain chemical or physical properties is the core issue of combinatorial chemistry. Since molecular topological indices are the topological invariants which can characterize the chemical or physical properties of molecules, it is much meaningful to investigate them. Different topological indices base on different parameters such as distance, eigenvalues and the number of matchings. Moreover some of them are of high computational complexity. So the optimization of topological indices is a rather difficult problem.With the comprehensive applications of molecular topological indices, more and more mathematicians and chemists concentrate on this area and obtain lots of meaningful new results. Among all topological indices, the Balaban index and Laplacian energy of graphs are two classes of widely-applied topological indices. Due to their fine function in characterizing the properties on chemistry or physics of molecules, many researchers in both mathematics and chemistry attach more attention to them. The content of this thesis is organized as follows.In Chapter 1, the background of the Balaban index and Laplacian energy of graphs is presented and main results in thesis are shown.In Chapter 2, the graphs with the maximal Balaban index are determined among all unicyclic graphs, bicyclic graphs and cacti, respectively.In Chapter 3, the weighted adjacency matrix and the weighted Laplacian matrix based on the Balaban index are introduced. Furthermore, several tight upper and lower bounds of the Balaban index are obtained by using their eigen-values. Combing with above two matrixes and the definition of the Balaban index, we introduce the higher Balaban index, and obtain some of its upper and lower bounds by using Cauchy-Schwarz inequality, the orthogonal decomposition of vectors and the properties of the eigenvalues of above two matrixes.In Chapter 4, we prove that the Laplacian energy of the n-vertex trees with perfect matchings, of diameter 4 or 5, is larger than that of the n-vertex path, which partly confirms a conjecture proposed by Radenkovic and Gutman in 2007 on the Laplacian energy. In addition, some comparative results between the Laplacian energy of the n-vertex trees with perfect matchings of diameter 4 and that of the n-vertex trees with perfect matchings of diameter 5 are shown as well.