Graphs Equienergetic With Edge-deleted Subgraphs And Estimate Of The Perron Root Of Nonnegative Matrices

The energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix, i.e. the sum of its singular values. The energy of a graph may increase, decrease or unchange when some edges are deleted. Our goal is to find a type of graphs whose energy unchanges when one edge is deleted.Nonnegative matrices appear in many areas naturally, such as, economics, social statis-tics. Nonnegative matrix theory is used widely in numerical analysis, graph theory, computer science and management science as an fundamental tool. Estimating the Perron root of non-negative matrices is one of the main problems in this field. If the bounds of the Perron root can be expressed as a function of the entries which is easy to compute, then the estimates are more useful. The main content of this thesis is as follows:(1) In Chapter 1, we introduce the relationship between simple graphs and matrices, and find a new type of equienergetic graph by studing the adjacency matrices.(2) In Chapter 2, we give a new estimate of the Perron root of nonnegative matrices which improves the existing bounds.