The Graphs Of Diameter N-4 With The Second Smallest Spectral Radius

The spectral of graphs is an important research subject in algebraic graph theory, which includes the adjacency spectrum and Laplacian spectrum of graphs, and so on. The tree is a very special and important graph. Based to this reason,many study for connected graphs often rely on trees.The paper will study the adjacency spectrum of graphs. At present there have been many good conclusions ordering graphs by the maximum adjacency spectrum radius of graphs, but there have been little conclusions ordering graphs by the minimum adjacency spectrum radius of graphs. On the basis of the previous conclusions, the purpose of this paper is to further determine the connected graph with the second smallest spectral radius in the set of all graphswith order n and diameter D∈n-2, n-3,n-4. The paper will be arranged as following: 1. The first chapter will introduce the general outline of the graph spectrum theory, related concepts and notations, and explain the structure of this paper.2. The second chapter will first order trees in the set of trees with order n and diameter n-2, then partition the trees with diameter n-3 and n-4 into some types, investigate their properties, order them in each type according to the minimum spectral radius, and determine the trees with the second smallest spectral radius in each type.3. The third chapter will prove that the graph with the second smallest spectral radius must be trees in the set of graphs with n vertices and diameter n-2, n-3, n-4.