A PROPERTY OF THE WIENER NUMBER AND ITS MODIFICATIONS

where S; and P; are the n-vertex star and the nvertex path, respectively (see Fig. 1). The bounds (6) have been independently established by several authors, and were probably first reported by Entringer et al.s They are now-a-days well known in chemical graph theory 7• We now demonstrate that the bounds (6) are special cases of some much more general results: Theorem 1. Let G be a connected acyclic graph, possessing n vertices. Let D be the diameter of G (= the maximum vertex distance in G). Let <lJ be the graph invariant defined via Eq. (5). If the function fix) monotonically increases for xE(l,D), then