On the Relation between the P′/P Index and the Wiener Number ² Dejan Plavs ˇic ´,* ,‡ Milan S ˇ os ˇkic ´, § Irena Landeka, ⊥ and Nenad Trinajstic ´ ‡ The Rugjer Bos ˇkovic ´ Institute, P.O.B. 1016, HR-10001 Zagreb, The Republic of Croatia, and Department of Chemistry, Faculty of Agriculture, and Department of Biochemistry, Faculty of Science, University of Zagreb, HR-10000 Zagreb, The Republic of Croatia Received May 6, 1996 X It is shown analytically that the recently introduced P ′/P index and the Wiener number are closely related graph-theoretical invariants for connected undirected acyclic structures and cycles. INTRODUCTION Molecular structure is the central theme of chemistry. 1-3 According to the principle of molecular structure, properties and behaviour of molecules follow from their structures. This statement has also been a subject of some criticisms. 4,5 If one considers nonmetric properties of a molecule, then the molecule can be represented by a (molecular) graph, which is essentially a nonnumerical mathematical object. Measur- able properties of a molecule are usually expressed by means of numbers. Hence, to correlate property or activity of a molecule with its topology, one must first convert by an algorithm the information contained in the graph to a numerical characteristic. A scalar numerical descriptor uniquely determined by a molecular graph is named a topological (graph-theoretical) index. 6,7 In the past the selection of graph matrices used for deriving of molecular indices was limited to the adjacency matrix A and the distance matrix D. 8,9 The situation has been changed in the last few years, and quite a few novel graph-theoretical matrices have been proposed. 10-15 Randic ´ has recently put forward a novel structure - explicit graph matrix P as well as the novel molecular index P′/P derived from it. 16,17 He also tested the new index by examining the octane numbers of octanes and empirically found a linear relationship between the P ′/P index and the Wiener number 18,19 in octanes. The representation of a molecule by a single number (topological index) entailes a considerable loss of information concerning the molecular structure. In search of new invariants which would improve the graph-theoretical char- acterization of molecular structure, a great number of indices have been proposed so far. 20,21 To make the evaluation of the existing and the future indices easier, Randic ´ put forward a list of desirable attributes for topological indices. 16 A particularly important requirement is that an index is not trivially related to, or highly intercorrelated with, other indices. If an index does not fulfill this condition then its informational content is either entirely or in major part comprised in other indices, as for instance in case of the Schultz index and the Wiener number 22,23 or the Hosoya Z index and the 1 Z index. 24 In this article we will discuss the relationship between the P′/P index and the Wiener number for connected undirected acyclic graphs and cycles. DEFINITIONS P Matrix. The P matrix of a labeled connected undirected graph G with N vertices, P ) P(G) is the square symmetric matrix of order N whose entry in the ith row and jth column is defined as where p′ ij is the total number of paths in the subgraph G′ obtained by the removal of the edge e ij from G, and p is the total number of paths in G. “Otherwise” means that either the vertices v i and v j are not adjacent or i ) j. If G′ is disjoint then the contributions of each component could be for instance, added 16 or multiplied. 25 Here we follow the Randic ´’s route, i.e., the addition of contributions. The P matrix can be the source of quite a few novel graph invariants-molecular indices. The process of finding P matrix is ilustrated for 2,3-dimethylpentane in Figure 1. P′/P Index. The P′/P index, P′/P ) P′/P(G), of a graph G is defined by means of the P matrix entries as The quantity p′ ij /p could be understood as a graphical bond order, 26 π e ij of the edge (bond) e ij of G. It is a measure of relative “importance” of an edge in a graph. Using the sum over all edges in G of these local quantities one obtains the graph invariantsmolecular index P ′/P. Wiener Number. The Wiener number, W ) W(G), of a graph G was introduced as the path number. 18 H. Wiener defined the path number as the number of bonds between all pairs of atoms in an acyclic molecule. Later W was defined in the framework of graph theory 19 by means of the ² Reported in part at the 1995 International Chemical Congress of Pacific Basin Societies (Pacifichem ‘95) - Frontiers in Mathematical Chemistry, Honolulu, Dec 17-22, 1995. ‡ The Rugjer Bos ˇkovic ´ Institute. §Department of Chemistry, Faculty of Agriculture. ⊥ Department of Biochemistry, Faculty of Science. X Abstract published in AdVance ACS Abstracts, November 1, 1996. (P) ij ) { p′ ij /p if the vertices v i and v j are adjacent in G 0 otherwise (1) P ′/P ) ∑ i)1 N-1 ∑ j>i N (P) ij (2) P ′/P ) ∑ e ij π e ij (3) 1123 J. Chem. Inf. Comput. Sci. 1996, 36, 1123-1126 S0095-2338(96)00369-1 CCC: $12.00 © 1996 American Chemical Society