Ordering trees with nearly perfect matchings by algebraic connectivity

Abstract

Let T _2k+1 be the set of trees on 2k+1 vertices with nearly perfect matchings and α(T) be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T _2k+1. Specifically, 10 trees T ₂,T ₃,⋯,T ₁₁ and two classes of trees T(1) and T(12) in T _2k+1 are introduced. It is shown in this paper that for each tree T ^′₁ , T ^″₁ ∈ T(1) and T ^′₁₂ , T ^″₁₂ ∈ T(12) and each i, j with 2 ≤ i < j <-11, α(T ^′₁ ) = α(T ^″₁ ) > α(T _i ) > α(T _j ) > α(T ^′₁₂ ) = α(T ^″₁₂ ). It is also shown that for each tree T with T ∈ T _2k+1 (T(1) ∪ {T ₂,_{T 3},⋯,T ₁₁} ∪ T(12)), α(T ^′₁₂ ) > α(T).