Hyper-Wiener vector, Wiener matrix sequence, and Wiener polynomial sequence of a graph
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The Wiener matrix and the hyper-Wiener number of a tree (acyclic structure) were first introduced by Randic [1]. Randic and Guo and colleagues [2, 3] further introduced the higher Wiener numbers of a tree that can be represented by a Wiener number sequence 1W, 2W, 3W,..., where 1W = W is the Wiener index, and k-1,2, kW = R is the hyper-Wiener number. Later the definition of hyper-Wiener number was extended by Klein et al. for application to any connected structure. In this study, the definition of higher Wiener numbers is extended to be applicable to any connected structure. The concepts of the Wiener vector and hyper-Wiener vector of a graph are introduced. Moreover, a matrix sequence W(1), W(2), W(3),..., called the Wiener matrix sequence (or distance matrix sequence), and their sum k=1,2, W(k) = W(H), called the hyper-Wiener matrix, are introduced, where W(1) = D is the distance matrix, and the sum of the entries of upper triangle of W(k) (resp. W(H)) is just equal to kW (resp. R). A Wiener polynomial sequence and a weighted hyper-Wiener polynomial of a graph are also introduced. 2006 Wiley Periodicals, Inc.