Majorization and distances in trees

We investigate distance vectors in trees and introduce a new center concept based on the notion of majorization and discuss relations to location theory. For a tree T and a vertex v ∈ T , we define the distance vector d ( v ,·) = ( d ( v , w ) w ∈ T ), where d ( v , w ) denotes the distance between v and a vertex w . We characterize whenever d ( u ,·) is weakly majorized by d ( v ,·) for adjacent vertices u and v . Moreover, we introduce a new center concept in trees, the majorization‐center, and relate this to known centers and the set of balance vertices. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(4), 251–257 2007