Abstract
Definitions of Wiener W,1 and hyper-Wiener R2 numbers are reanalyzed and defined from a matrix-theoretical point of view. Thus, D and W1 (distance and Wiener,3,4 of paths of length 1) matrices are recognized as a basis for calculating W, whereas DP and WP (distance-path [this work] and Wiener-path,4 of paths of any length) are recognized as a basis for the calculation of R. Weighted walk degrees eWM,i generated by an iterative additive algorithm5 are considered as local vertex invariants (LOVIs) whose half-sum in graph offers walk numbers eWM which are Wiener-type numbers of rank e; for e = 1, the classical W and R numbers are obtained. New matrix invariants, Δ, DP (“combinatorial” matrices constructed on D), K (of reciprocal [DP]ij entries), and WU (of unsymmetrical weighted distance) are proposed as a basis for weighting walk degrees and whence for devising novel numbers of Wiener-type.
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