WIENER MATRIX - SOURCE OF NOVEL GRAPH INVARIANTS

We report some properties of new graph matrices which appear to offer novel graph invariants of potential interest in structure-property studies. The matrices are constructed by generalizing Wiener's procedure for evaluation of Wiener numbers in alkanes. Among the invariants considered we particularly examined the sequences generated by summing the entries in the matrix for vertices at the same distance from one another. These numbers may be viewed as ''higher'' Wiener numbers in analogy with ''higher'' connectivity indices. We have listed the higher Wiener numbers of alkanes up to n = 9 carbon atoms and also report several recursions for the construction of these invariants for selected families of acyclic graphs. Briefly, we have outlined how the Wiener matrix can be extended to cyclic systems, while in the concluding comments we have outlined an extension of the Wiener matrix to molecules having heteroatoms. The significance of the matrices as a source of graph invariants is precisely in this possibility to go beyond simple models of molecular graphs and extend graph invariants of interest to molecules having different kinds of atoms.