On majorization of closed walk vectors of trees with given degree sequences

Let Cv(k; T) be the number of closed walks of length k starting at vertex v in a tree T. We prove that for any tree T with a given degree sequence π, the vector C(k; T) ≡ (Cv(k; T), v ∈ V(T)) is weakly majorized by the vector C(k;Tπ*)≡(Cv(k;Tπ*),v∈V(Tπ*)), where Tπ* is the greedy tree with the degree sequence π. In addition, for two trees degree sequences π and π′, if π is majorized by π′, then C(k;Tπ*) is weakly majorized by C(k;Tπ′*).