Collect. Czech. Chem. Commun. 1989, 54, 2148-2155

Path and walk matrices of trees

Milan Kunz

Chemopetrol, Research Institute of Macromolecular Chemistry, 656 49 Brno


The relation between the Rouse matrix R and the Zimm matrix Z, defined for polymer chains, is generalized for all trees. The path matrix Wu of an unoriented tree and the walk matrix Wo of an oriented tree are defined and their relations with the corresponding incidence matrices G and S of trees: (GGT) = n(WuTWu)-1 and (SST) = n(WoTWo)-1 are proved. The elements of the quadratical forms WTW are the number of path (selfavoiding walks) in which the edge (arc) i is present together with the edge (arc) j. The traces of WTW are the Wiener path number.