On The Second Laplacian Eigenvalue Of The Connected Graph

The theory of the graph spectra is an important branch in graph theory, there are many important applications in the fields of physics, chemistry, computer science and other fields.There is a natural connection between the theory of the graph spectra and the structure of the graph. The theory of the graph spectra can reflect the nature of graph theory. Therefore graph spectra is paid more and more attention by people. It is a hot issue in current study of graph theory.The study on the Laplacian spectrum of graphs has three kinds of methods: algebraic method, geometric method and probability method. In this paper, we study the second largest Laplacian eigenvalue of the graph by using the method of algebra and geometry.The main contents are as follows:1. In this paper, it introduces the background and significance of Laplacian eigenvalue and the graph theory, and gives the concept of symbols and research status at home and abroad about Laplacian eigenvalue.2. In this paper, it summarizes the related basic theories and the influence diagram operations on the Laplacian eigenvalue of the graph.3. In this paper, it respectively studied the graph including circle and tree that their second largest Laplacian eigenvalue no more than 2 ?2. Finally, in the case of n ?7, it gave all the connected graph of the second largest Laplacian eigenvalue no more than 2 ?2.