A topological interpretation of the walk distances

Abstract

The walk distances in graphs have no direct interpretation in terms of walk weights, since they are introduced via the logarithmsof walk weights. Only in the limiting cases where the logarithms vanish such representations follow straightforwardly. The interpretation proposed in this chapter rests on the identity applied to the cofactors of the matrix where Ais the weighted adjacency matrix of a weighted multigraph and tis a sufficiently small positive parameter. In addition, this interpretation is based on the power series expansion of the logarithm of a matrix. Kasteleyn (Graph theory and crystal physics. In: Harary, F. (ed.) Graph Theory and Theoretical Physics. Academic Press, London, 1967) was probably the first to apply the foregoing approach to expanding the determinant of I- A. We show that using a certain linear transformation the same approach can be extended to the cofactors of I- tA, which provides a topological interpretation of the walk distances.