J. Chem. Inf. Comput. Sci. 1996, 36, 535-540
535
Walk Numbers eWM: Wiener-Type Numbers of Higher Rank† Mircea V. Diudea Department of Chemistry, “Babes-Bolyai” University, Arany Janos Str. 11, 3400 Cluj, Romania Received September 30, 1995X
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, ∆, 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. INTRODUCTION
Wiener1
has defined his number W as “the sum of the distances between any two carbon atoms in the molecule, in terms of carbon-carbon bonds”. W number (or the path number) can be calculated1 as the sum of bond contributions We of all edges, in an acyclic graph, G
W ) W(G) ) ∑eWe ) ∑eNL,e·NR,e
(1)
NL,e + NR,e ) N(G)
(2)
with
NL, NR being the number of vertices lying to the left ant to the right of edge e, and the summation runs over all edges in G. Lukovits6-8 extended the “bond contribution” definition (or method of calculation) for cycle-containing graphs, giving for We the following relation
We ) ∑Ceij/Cij
(3)
i
