The third smallest eigenvalue of the Laplacian matrix (Q2736391)

It is well known that the second smallest eigenvalue of the Laplacian of a graph bears a lot of information about combinatorial properties of the graph. In this paper, the relationship between the third smallest eigenvalue of the Laplacian matrix and the graph structure is explored. It is assumed that the graph is connected and simple, and that the second eigenvalue has multiplicity one. Under these assumptions, the sign structure of the eigenvector corresponding to the third eigenvalue is described in the case when the graph is a tree, and some results about the multiplicity of this eigenvalue are given.