The Laplace Spectrum Of A Single Circle In Figure

The Laplace matrix is an active and important subject in Algebra Graph Theory. It is a discrete form of the Laplace operator on the compact Rimannian manifold acting on the graphs. It has been widely used in many fields such as Physics, Chemistry, Biology, Computer net science and Information theory ,etc. Some results on the Laplace matrix are given in this thesis.Applying the De Caen's inequality of sum of the squares of the degree and Cauchy's inequality, we obtain a strict lower bound and a strict upper bound of the largest Laplace eigenvalues only in terms of vertex number of a unicycle graph. Applying the Laplace matrix theorem of trees, we obtain an upper bound of the second smallest Laplace eigenvalues of a unicycle. Extremal graph whose second smallest Laplace eigenvalues reach the obtained upper bound is determined. We also obtain an upper bound of the second largest Laplace eigenvalues in terms of vertex number of the largest connected branch of unicycle graph, and obtain a theoretical method to calculate the second largest Laplace eigenvalues of unicycle graph. We obtain an upper bound of any Laplace eigenvalues in terms of vertex number of a unicycle graph. We also obtain the distribution of Laplace eigenvalues in the inter [0,n] in terms of the matching number.Applying the nonnegative matrix theorem, we obtain a strict lower bound of the Q-spectrum of unicycle graph. In addition, we obtain an upper bound of the largest Laplace eigenvalues in terms of the Largest Q-eigen values. At last, we obtain the necessary and sufficient condition for judging the equality between Q-spectrum and Laplace spectrum.We obtain an upper bound of the largest Laplace eigenvalues of double-cycle graph by using its vertex number. Besides, the relation between the Laplace eigenvalues of unicycle and double-cycle is presented by the generating relation between them.