The Second Zagreb Indices and Wiener Polarity Indices of Trees with Given Degree Sequences

Given a tree T = (V,E), the second Zagreb index of T is denoted by M2(T ) = ∑ uv∈E d(u)d(v) and the Wiener polarity index of T is equal to WP (T ) = ∑ uv∈E(d(u)−1)(d(v)−1). Let π = (d1, d2, ..., dn) and π′ = (d1, d2, ..., dn) be two different non-increasing tree degree sequences. We write π π′, if and only if ∑n i=1 di = ∑n i=1 d ′ i, and ∑j i=1 di ≤ ∑j i=1 d ′ i for all j = 1, 2, ..., n. Let Γ(π) be the class of connected graphs with degree sequence π. In this paper, we characterize one of many trees that achieve the maximum second Zagreb index and maximum Wiener polarity index in the class of trees with given degree sequence, respectively. Moreover, we prove that if π π′, T ∗ and T ∗∗ have the maximum second Zagreb indices in Γ(π) and Γ(π′), respectively, then M2(T ∗) < M2(T ∗∗).