Centers to centroids in graphs

For S contained in V(G) the S-center and S-centroid of G are defined as the collection of vertices u an element of V(G) that minimize e/sub s/(u) = max (d(u,v): v an element of S) and d/sub s/(u) = ..sigma../sub v an element of S/d(u,v), respectively. This procedure generalizes the standard definition of center and centroid from the special case of S = V(G). For 1 less than or equal to k less than or equal to absolute value V(G) and u an element of V(G) let r/sub k/(u) = max (..sigma../sub s an element of S/d(u,s): S contained in V(G), absolute value S = k). The k-centrum of G, denoted C(G;k), is defined to be the subset of vertices u in G for which r/sub k/(u) is a minimum. This approach also generalizes the standard definitions of center and centroid since C(G;1) is the center of C(G; absolute value V(G)) is the centroid. The structure of these sets for trees is examined. Generalizations of theorems of Jordan and Zelinka are included. 4 figures, 2 tables.