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 M-2(T) = Sigma(uv is an element of E) d(u)d(v) and the Wiener polarity index of T is equal to W-P(T) = Sigma(uv is an element of E)(d(u)-1)(d(v)-1). Let pi = (d(1), d(2),..., d(n)) and pi' = (d(1)', d(2)',..., d(n)') be two different non-increasing tree degree sequences. We write pi (sic) pi', if and only if Sigma(n)(i=1) d(i) = Sigma(n)(i=1) d(i)', and Sigma(j)(i=1) d(i) <= Sigma(j)(i=1) d(i)' for all j = 1, 2,..., n. Let Gamma(pi) be the class of connected graphs with degree sequence pi. 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 pi (sic) pi', T* and T** have the maximum second Zagreb indices in Gamma(pi) and Gamma(pi'), respectively, then M-2(T*) < M-2(T**).