J. Chem. Inf. Comput. Sci. 1997, 37, 349-354
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Trees with Extremal Hyper-Wiener Index: Mathematical Basis and Chemical Applications Ivan Gutman,†,| Wolfgang Linert,*,‡ Istva´n Lukovits,§ and Andrey A. Dobrynin⊥ Institute for Advanced Studies, The Hebrew University of Jerusalem, Jerusalem, Israel, Vienna University of Technology, Institute of Inorganic Chemistry, Getreidemarkt 9, A-1060 Vienna, Austria, Central Research Institute for Chemistry, Hungarian Academy of Sciences, H-1525 Budapest, P.O. Box 17, Hungary, and Institute of Mathematics, Russian Academy of Sciences, Siberian Branch, Novosibirsk 630090, Russia Received September 27, 1996X
Trees with minimal and maximal hyper-Wiener indices (WW) are determined: Among n-vertex trees, minimum and maximum WW is achieved for the star-graph (Sn) and the path-graph (Pn), respectively. Since WW(Sn) is a quadratic polynomial in n,, whereas WW(Pn) is a quartic polynomial in n, the hyper-Wiener indices of all n-vertex trees assume values from a relatively narrow interval. Consequently, the hyperWiener index must have a very low isomer-discriminating power. This conclusion is corroborated by finding large families of trees, all members of which have equal WW-values. INTRODUCTION
The hyper-Wiener index (WW) is a graph invariant, put forward by Randic´ a few years ago1 as a kind of extension of the classical Wiener-index-concept. Randic´’s article1 stimulated a large number of subsequent researches2-10 on WW, and the main properties of this topological index are nowadays relatively well understood. Randic´’s starting point was the following long-known11,12 property of the Wiener index (W) of trees. Let T be a tree (i.e., a connected and acyclic graph). Let e be an edge of T, joining the (adjacent) vertices u and V. Denote by nu(e) the number of vertices of T lying on one side of the edge e, closer to vertex u. Denote by nV(e) the number of vertices of T lying on the other side of the edge e, closer to vertex V. (Because T is acyclic, the quantities nu(e) and nV(e) are unambiguously determined for every edge.) Then the Wiener index of T satisfies the relation11
W(T) ) ∑nu(e)nV(e) e
in which the summation goes over all edges of T. Now, instead of adjacent vertices, Randic´ extended the consideration to arbitrary pairs of vertices u,V of the tree T. These vertices are joined by a unique path which we denote by p. Denote by nu(p) the number of vertices of T lying on one side of the path p, closer to vertex u. Denote by nV(p) the number of vertices of T lying on the other side of the path p, closer to vertex V. (Again, because T is acyclic, the quantities nu(p) and nV(p) are unambiguously determined for every path.) Then the hyper-Wiener index of T is defined by means of the relation1 †
The Hebrew University of Jerusalem. Vienna University of Technology. Hungarian Academy of Sciences. ⊥ Russian Academy of Sciences. | On leave from the Faculty of Science, University of Kragujevac, P.O. Box 60, YU-34000 Kragujevac, Yugoslavia. X Abstract published in AdVance ACS Abstracts, February 15, 1997. ‡ §
S0095-2338(96)00139-4 CCC: $14.00
WW(T) ) ∑nu(p)nV(p)
(1)
p
in which the summation goes over all paths of T. As well known, the Wiener index is defined as the sum of distances between all pairs of vertices of the respective graph:
W(G) ) ∑d(x,y|G) x
