On The Map Laplace Spectrum And The Adjacency Spectrum Of Research

The spectral theory of graph is not only an important field in graph theory but also an active topic. There are extensive applications in the fields of quantum chemistry, physics, computer science, communication network and information science. The theory of graphs also have relation closely with the fixed variate (e.g. chromatic number, degree sequence, diameter and connectivity) of graphs. The spectral theory of graph mainly include the adjacency spectrum,Laplace-spectrum,Q-spectrum,C-spectrum,S-spectrum,which adjacency spectrum and Laplace-spectrum are most universal. In many applications, good upper bounds for the largest Laplacian eigenvalues of a graph G are needed.In this thesis, there are three important aspects in the spectral theory of graph of the simple connected and undirected graph, one is the Laplace-spectrum, another is the adjacency spectrum. Last is the relations of these spectrums. Mainly divided four parts,we mainly study the simple connected undirected graph and give the following conclusion:1,By separating the matrix into two matrices and using famous Wely theorem, give the relation about the adjacency spectrum of a graph and its line graph. This conclusion promotes the old result which the relation about the adjacency spectrum radius of a graph and its line graph to all the eigenvalues. In addition, using the same method, have the relation about the adjacency spectrum of a graph and its Laplace-spectrum.2,Using the knowledge, which the number of walks of length 2 in the graph G with an end at v is equal the sum of degree of u over all u with uv∈E, have two new bounds of the simple connected undirected graph's largest Laplacian eigenvalues, which is more precise to some graphs. In addition, using the covering number of G , also have two new bounds of the largest Laplacian eigenvalues.3,In this thesis give two new upper bounds on the sum of Laplace spectral radii(spectral radii of the Nordhaus–Gaddum type)of a simple graph and its complement by means of algebraic method, the condition of maximum and minimum of a function. Through some examples explained our results are more precise than formerly results. In addition, also have for the difference of Laplace spectral radii and the algebraic connectivity and also attained a new upper bound for the product of Laplace spectral radii of a graph and its complement.