The six classes of trees with the largest algebraic connectivity

Abstract

In this paper, we study the algebraic connectivity α ( T ) of a tree T. We introduce six Classes ( C 1 ) – ( C 6 ) of trees of order n, and prove that if T is a tree of order n ⩾ 15 , then α ( T ) ⩾ 2 - 3 if and only if T ∈ ⋃ i = 1 6 Ci , where the equality holds if and only if T is a tree in the Class ( C 6 ) . 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 α ( T i ) > α ( T j ) if 1 ⩽ i < j ⩽ 6 and T i is any tree in the Class ( Ci ) and T j is any tree in the Class ( Cj ) . 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.