Characterization For Q-spectrum Of Graph And The Numerical Invariants Associated With The Q-matrix And The Laplacian Matrix Of The Graph L_n,p

Graph spectral theory have a wide range of application in Physics, quantum chemistry, computer science, communication networks and information sciences and many others fields. The Laplacian spectrum and the signless Laplacian spectrum of a graph is an important part of the spectral of graph theory. The main research content of the relevant is: First, the use of the Laplacian matrix and the signless Laplacian matrix to study the structural properties of graphs. Second,study the numerical invariant associated with the Laplacian matrix and the signless Laplacian matrix. In recent years, there is a common numerical invariants,such as the Laplacian spectrum, Q-spectrum, algebraic connectivity of graph,Kirchhoff index, Q-matrix coefficients and so on. Let K p be a complete graph of order n, and let L n,p be the graph with n vertices obtained by identifying the vertex u i of K p with the vertex v i of the tree T i, where i = 1, 2,..., r and1 ≤ r ≤ p. This paper studies the Laplacian matrix and the signless Laplacian matrix of graphs, and the content are two relevant research topics: First, the characterization of a graph i.e. determining the graph with the same spectrum.Second, the extremal problem about the numerical invariants associated with the Q-matrix and the Laplacian matrix of a graph i.e. to determine the graph with the minimal or the maximal numerical invariants in given graphs. The main results as follow:1. Characterize the features of graphs which are satisfied with ν4< 2, and obtained all connected graph with ν4< 2, where ν4is the fourth large eigenvalue;2. Study the Algebraic connectivity of the graphs L n,p, and determined the graphs with the maximal, minimal and the second small Algebraic connectivity, respectively;3. Study the Kirchhoff index of the graphs L n,p, and determined the graphs with the maximal and the five small Kirchhoff index, respectively;4. Study the Q-characteristic polynomial coefficients of the graphs L n,p, and determined the graphs with the maximal and the minimal coefficients ζi(1 ≤i ≤ n).