Upper bounds on the smallest positive eigenvalue of trees with at most one zero eigenvalue

Abstract

In this article, we consider the trees for which the adjacency matrix has nullity at most one. We study the smallest positive (adjacency) eigenvalue of these trees. For such trees lower bounds on the smallest positive eigenvalue have already been obtained in literature but the problem of finding an upper bound still remains unsolved. In this article, we find upper bounds on the smallest positive eigenvalue of these trees and obtain corresponding unique maximal trees. Additionally, we study the smallest positive eigenvalue of trees obtained by attaching a pendant at some vertex of a path graph and also identify the trees for the extremal cases. As applications of our earlier results, we show that, for a path of odd order, the smallest positive eigenvalue is maximum, if we add the pendant almost in the middle. As another application, we show that for a path of any order, the smallest positive eigenvalue is minimum, if we add the pendant vertex at the end.