The Least Eigenvalue Of A Graph With Quasi-pendant Vertices

Spectral graph theory is an important branch of the alge-braic graph theory. We characterize the structural properties of graphs through the relation between the spectrum of graphs and the other parameters. It mainly involves the adjacency spectral, the Laplacian spectral and the signless Laplacian spectral. We study some of the extreme eigenvalues of graphs, such as the spec-tral radius, the least eigenvalue and the algebraic connectivity. Recently, the least eigenvalue problem receives much attention of the scholars.This thesis focuses on the least eigenvalue of the adjacency matrix. The early research was about the bounds. Because there is no appropriate methods, one can not get a breakthrough. In2008, Bell and Fan began the research about the minimizing graphs with some given parameters. Bell established the com-bination of eigenvectors method and Fan discovered the pertur-bation lemma.In chapter1, we introduce the research background, some concepts and notations, the problem and the main results we ob-tain in this thesis. In chapter2, we summarize some classical results of the least eigenvalue of adjacency matrix of graphs ac-cording to the research process, and then analyze the important methods and conclusions. Based on these methods and results, we investigate the least eigenvalue of the adjacency matrix of the graph with given quasi-pendant vertices in the third chapter. Fur-thermore we characterize the structure of the minimizing graph.