On trees with maximum algebraic connectivity

Abstract

In this paper, trees with fixed diameter and any number of vertices are investigated. A subclass of trees with diameter 2k is introduced, the diameter path trees (dp-trees). Two subclasses of dp-trees are defined in which we characterize the elements that maximize the algebraic connectivity. Also, it is proved that if any tree maximizes the algebraic connectivity over all trees with diameter 2k then it is a dp-tree. For such trees, a bound for the degrees of their vertices is given. In the case of the odd diameter, 2k - 1, we show that P2k is the only tree that maximizes the algebraic connectivity.