On Some Problems Of Spectrum Of A Graph

In this thesis, some problems of spectrum of graphs are studied. Firstly, the minimum rank among all symmetric and skew symmetric real matrices respectively of k-th power of trees are studied. A class of unicyclic graph is proved to be DS graph (graph determined by its spectrum) with respected to its Laplacian matrix. The bicyclic digraphs with extremal skew energy, which is defined to be the sum of absolute values of eigenvalues of its skew adjacency matrix, are determined.The main results are given in the following.1. In Chapter3, the minimum rank of k-th power of a tree is charac-terized partly.In paper [26], R. Brualdi, L. Hogben, B. Shader conjectured:If T≠K1,n-1, then mr(T3)≤mr(T2)-1, where K1,n-1is a star on n vertices. In this chapter, we point out that the conjecture is not always true. Some coun-terexamples are given. By matrix construction and zero forcing set, we also obtain the minimum rank of T2and the relation between mr(T2) and mr(T3) if T contains no2-vertex. The minimum ranks of k-th power of caterpillars are also characterized thoroughly in this chapter. The minimum rank of skew-symmetric matrices described by the k-th power of a caterpillar is studied.2. Let Gr,p be a graph obtained from a path by adjoining a cycle Cr of length r to one end and the central vertex of a star Sp on p vertices to the other end. In Chapter4, we prove that unicyclic graph Gr,p with r even is determined by its Laplacian spectrum except for n=p+4.3.Let G be a digraph and S(G) be the skew-adjacency matrix of G. The skew energy of G is the sum of the absolute values of eigenvalues of S(G). In Chapter5, the bicyclic digraphs with minimal and maximal skew energy are determined.