Characterization of the Number of Main Eigenvalues of a Class of Trees

摘要

The eigenvalues of the adjacency matrix of a graph are called the eigenvalues of a graph, the multi-set of all eigenvalues of a graph is called the spectrum of a graph. An eigenvalue of a graph is a main eigenvalue if its eigenspace is not orthogonal to the all-ones vector, the main eigenvalues are significant to the characterization of graphs and the properties of graphs. Characterizing graphs with k(2≤k≤n) number of main eigenvalues are a long-standing problem. In this paper, the lower bound of main eigenvalues of a class of trees is determined.