The Wiener Index and the Szeged Index of Benzenoid Systems in Linear Time

Distance properties of molecular graphs form an important topic in chemical graph theory.1 To justify this statement just recall the famous Wiener index which is also known as the Wiener number. This index is the first2 but also one of the most important topological indices of chemical graphs. Its research is still very active; see recent reviews3,4 and several new results in a volume5 dedicated to the 50th anniversary of Wiener’s paper.2 Many methods and algorithms for computing the Wiener index of a graph were proposed in the chemical literature; papers in refs 4, 6-10 present just a sample of these studies. The fastest general algorithm7 for computing the Wiener index of a graph is of complexity O(mn), where n and m denote the number of vertices and the number of edges, respectively, of a considered graph. Linear algorithms were previously proposed only for trees7 and for fasciagraphs and rotagraphs.9 We wish to add, however, that in the latter case the linearity is a more theoretical than practical term. This is due to the fact that the size of the problem is supposed to be the number of copies of a given graph G from which a fasciagraph (rotagraph) is build up and not the size of G itself. The Szeged index (Sz) of a (molecular) graph is a recently proposed11,12 topological index which has already received considerable attention.13-20 Among others, a method for its computation was given,14 and an O(mn) algorithm for its computation has been proposed.17 A new approach to the study of distance properties of molecular graphs was proposed by Klavžar, Gutman, and Mohar.21 It was shown that benzenoid systems provide socalled isometric embeddings into hypercubes, and based on this fact a simple formula for the Wiener index of these graphs has been obtained. The approach was further developed in the subsequent papers.10,14 In particular, combinatorial expressions for the Wiener index of compact pericondensed benzenoid hydrocarbons were computed.10 Along these lines it was observed22 that benzenoid systems can also be isometrically embedded into the Cartesian product of three trees. As an application it was demonstrated how to compute the diameter of a benzenoid system in linear time. Here we continue to develop this line of research. Our main result asserts that the Wiener index of a benzenoid system with n vertices can be computed in O(n) time. In the next section we present the algorithm and prove its correctness. We also give an example from which it can be seen that this linear algorithm for the Wiener index not only is fast on computers but also is conceptually simple and easy to follow on the paper. In the last section we demonstrate that our approach can also be used to compute the Szeged index of a benzenoid system on n vertices in O(n) time. The Wiener index or Wiener number of a (molecular) graph G ) (V,E) is defined as follows: