Types Of Graph Spectrum

In this paper, we mainly study the spectrum of graphs, including the adjacency spectrum and the Laplacian spectrum. Some results will be given in this thesis. 1. We first present a sharp bound for the Laplacian spectral radius as follows:where 1 < i < n. When i = 2 the equality holds if and only if G is a regular bipartite graph, when i = 2 the equality holds if and only if G is a regular bipartite graph or a star graph.2. The algebraic connectivity of trees is determined by the graft transformation, meanwhile the relation between the algebraic connectivity and the diameter of two classes of trees is also determined.3. We also give the upper and low bounds for the spectral raduis of graphs whose vertex connectivity is k.4. Finally, we investigate the largest and smallest eigenvalues of the double cyclic. Moreover, we characterize all extremal graphs with these bounds.