Main Eigenvalue Of Graph And Equitable Partition

The spectral graph theory is an important branch of algebraic graph theory, mainly including adjacency spectrum, Laplacian spectrum and Signless-Laplacian spectrum et-al, and the former originated with the quantum chemistry. However, the main work of the spectral graph theory is to study the spectra of multiform matrices of graphs and discuss the relations between the spectra and graph invariants, structure properties of graphs by means of the matrix theory, combinatorics theory. For a graph G, the eigenvalue A of A{G) is said to be a main eigenvalue if the eigenspace(A) is not orthogonal to all one vector j, and non-main otherwise. A long standing problem posed by Cvekovic [1] is that how to characterize graphs with s (s≥2)main eigenvalues. In this article, we mainly study to the relation between main partition and equitation partition. In addition, we focus to give a class of graphs that exactly have two (three) main eigenvalues. The main results are as follows:In the first chapter, we introduce some basic definitions, symbols and notations about main eigenvalue, and recalled the history, and list some important known results about the main eigenvalues of graph.The second chapter consists of two sections. In the first section, we cite some important lemmas, corollaries and theorems. In the second section, we introduce an important definition about k-main partition of graph. We get some results about main eigenvalues of graph using k-main partition and equitable partition properties.The third chapter consists of three sections. In the first section, we give a parameter table which built a relation between s-main eigenvalues and equitable partitions. In the second section, we mainly focus to some classes of graphs have two main eigenvalue. Especially, we mainly characterize a class of graph has two main eigenvalue, and have equitable k-partition for any even positive integer. In the third section, we mainly focus to characterize some classes of graphs which have three main eigenvalues.