| Let G be a simple undirected graph on n vertices. In this thesis we study the minimum rank m(G) of real, symmetric matrices with prescribed graph G, and matrices which attain the attendant minimum. We concentrate on the case where the graph has no cut-vertex. A more general case is also considered. We also devote some effort to properties of a matrix B = D − A(G) which is associated with a graph, where D is a diagonal matrix and A (G) is the well-know (0,1) adjacency matrix of a graph G. In the last of this thesis, we discuss a positive integer multiset {lcub}s1,s2 ,…,sk{rcub} which we called a multiplicity set of a graph G, where si is an algebraic multiplicity of an eigenvalue of a matrix associated with G. |
