The Chromatic Number And Estrada Index Of Graphs

Spectral graph theory mainly discuss relations between the structural property and the spectral property of graphs by using the spectra of graphs. An important application of spectral graph theory in structural chemistry is that one can characterize the chemical or physical properties of chemical molecule structure by using the spectra of graphs corre-sponding to chemical molecule structure. So, researchers propose some molecular topological index based on spectrum of graphs.In2000, an index was introduced by E.Estrada to measure the degree of folding of a protein, which was called the Estrada index by other researchers. In recent years, the Estrada index have became a hot topic of spectral graph theory, have received much attention. In this dissertation, we survey the recent development of Estrada index from two aspects, discuss the relationship between the Estrada index and the chromatic number which is an important graph parameter, characterize the graph whose Estrada index attains maximum among all graphs with fixed order and given chromatic number, and partially characterize the minimizing graph in this graph set.The dissertation is organized as follows. In Charter one, we briefly introduce the back-ground of the Estrada index and some concepts and notations which will be used in following Charter. In Charter two, the recent development of the Estrada index is surveyed from two aspects. In final Charter, we characterize the maximizing graph among all graphs with fixed order and given chromatic number, and characterize the minimizing graph among all graphs whose chromatic number is one less than its order.