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

Let C v ( 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 ) ≡ ( C v ( k ; T ), v  ∈  V ( T )) is weakly majorized by the vector C ( k ; T π * ) ≡ ( C v ( k ; T π * ) , v ∈ V ( T π * ) ) , where T π * 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 ; T π * ) is weakly majorized by C ( k ; T π ′ * ) . [ABSTRACT FROM AUTHOR]