Ordering trees with algebraic connectivity and diameter

This paper is motivated by Abreu’s paper which proposed “finding a total ordering of trees by their algebraic connectivity is still an open problem.” We first discuss how to find a total ordering of trees with diameter 4. Then these results are used to determine first several orderings of trees by algebraic connectivity in nonincreasing order. In last, we present a class of trees with large diameter whose algebraic connectivity is larger than that of a tree with diameter 3.