About The Fitting Laplacian Eigenvalue Of A Graph

Spectral graph theory as one of the main research directions of the algebraic graphtheory, has a long history. However, Its extensive research originated in the early1970s.It mainly includes the adjacent spectral theory and Laplacian spectral theory. Laplacianspectral theory is currently not only a hot research direction of Spectral graph theory,but also one of the research directions of Combined matrix theory. In recent years, thesignless Laplacian of graphs has attracted the attention of some researchers. Since thespectrum of signless Laplacian has importance in graph theory, matrix theory and thedefinite solution of partial differential equations, also has applicability in quantumchemistry, biology and complex network. It is widespread concern in recent research ofSpectral graph theory. Therefore, It has important theoretical and practical values tostudy the eigenvalues of signless Laplacian.In this paper, we study the eigenvalues of signless Laplacian by means of matrixtheory, algebraic methods and graph operations. Main work as follows:1. We give a brief introduction to the concept, research background and methods ofthe Q-theory and summarize some basic properties of Q-spectrum, in particular, westudy the Q-spectral properties of regular graphs, bipartite graphs and Starlike trees. Inaddition, we generalize the basic results of graph operations on Q-spectrum.2. We summarize some basic properties and results of Q-spectral radiiq1andprove an upper bound by using a new method. Moreover, an upper bound of Starliketrees is obtained.3. We obtain a constant lower bound of Q-spread and get some corollariesincluding lower bound for Q-spread of regular graphs, bipartite graphs and a lowerbound for Q-spectral radii of connected graphs.4. An upper bound for signless Laplacian energy is obtained by using Cauchy-Schwarz inequality. Meanwhile, we get some basic properties of signless Laplacianenergy by studying the line graph.