The six classes of trees with the largest algebraic connectivity Original Research Article

In this paper, we study the algebraic connectivity image of a tree T. We introduce six Classes image–image of trees of order n, and prove that if T is a tree of order image, then image if and only if image, where the equality holds if and only if T is a tree in the Class image. At the same time we give a complete ordering of the trees in these six classes by their algebraic connectivity. In particular, we show that image if image and image is any tree in the Class image and image is any tree in the Class image. We also give the values of the algebraic connectivity of the trees in these six classes. As a technique used in the proofs of the above mentioned results, we also give a complete characterization of the equality case of a well-known relation between the algebraic connectivity of a tree T and the Perron value of the bottleneck matrix of a Perron branch of T.