Walk numbers W-e(M): Wiener-type numbers of higher rank

Definitions of Wiener W,(1) and hyper-Wiener R(2) numbers are reanalyzed and defined from a matrix-theoretical point of view. Thus, D and W-1 (distance and Wiener,(3,4) of paths of length 1) matrices are recognized as a basis for calculating W, whereas D-p and W-p (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 W-e(M,i) generated by an iterative additive algorithm(5) are considered as local vertex invariants (LOVIs) whose half-sum in graph offers walk numbers W-e(M), which are Wiener-type numbers of rank e; for e = 1, the classical W acid R numbers are obtained. New matrix invariants, Delta, D-p (''combinatorial'' matrices constructed on D), K (of reciprocal [D-P](ij) entries), and W-U (of unsymmetrical weighted distance) are proposed as a basis for weighting walk degrees and whence for devising novel numbers of Wiener-type.