Centers to centroids in graphs

For S ⊆ V(G) the S‐center and S‐centroid of G are defined as the collection of vertices u ∈ V(G) that minimize es(u) = max {d(u, v): v ∈ S} and ds(u) = ∑u∈S d(u, v), respectively. This generalizes the standard definition of center and centroid from the special case of S = V(G). For 1 ⩽ k ⩽|V(G)| and u ∈ V(G) let rk(u) = max {∑s∈S d(u, s): S ⊆ V(G), |S| = k}. The k‐centrum of G, denoted C(G; k), is defined to be the subset of vertices u in G for which rk(u) is a minimum. This also generalizes the standard definitions of center and centroid since C(G; 1) is the center and C(G; |V(G)|) is the centroid. In this paper the structure of these sets for trees is examined. Generalizations of theorems of Jordan and Zelinka are included.