On The Least Q-eigenvalue Of Some Non-bipartite Graphs

In order to study the structure and property of a graphs, people introduce various matrices, the main of which are adjacency matrix, Laplacian matrix and signless Laplacian matrix. People have carried out extensive research for the eigenvalues of the adjacency matrix and Laplacian matrix, and have obtained lots of mature theory and application results. Until recently ten years, researches have found that the signless Laplacian matrix of a graph is more closely related to the structure of the graph, and more convenient to study the properties of the graph by the signless Laplacian matrix.The signless Laplacian matrix of a graph is also called Q-matrix of the graph,the eigenvalues of which is called Q-eigenvalues of the graph. For a connected graph,the least Q-eigenvalue is zero if and only if the graph is bipartite. And the least Q-eigenvalue of a graph is even used to measure of non-bipartiteness of the graph. It has attracted the attention of many scholars, which has become a hot issue in the study of spectra graph theory.For a class of non-bipartite connected graphs, determining the graphs whose least Q-eigenvalue attains the minimum among all the graphs in the class is wildly researched recently. People have determined the graphs whose least Q-eigenvalue attains the minimum for many classes of non-bipartite connected graphs. This paper further studies the problem and the main contents are as follows:The first chapter mainly introduces the background and the main progress of the least Q-eigenvalue of a graph, and overview the main results obtained in this paper.The second chapter introduces some notations, concepts and lemmas, and proves some new lemmas.The third chapter studies conditions depending on the least Q-eigenvalue of a graph under which the graph contains a long path, and determine the extremal graph in which the least signless Laplacian eigenvalue attains the minimum among all the Pt-free non-bipartite unicyclic graphs and Pt-free non-bipartite connected graphs of order n, respectively.The forth chapter studies the least Q-eigenvalue of non-bipartite unicyclic graphs of fixed order and number of pendant vertices, and determines the graph whose least Q-eigenvalue attains the second smallest. For the graphs whose least Q-eigenvalue attains the second and third smallest, we give some theorems and conjectures.