A new formalism for the study of Wiener numbers is developed which makes use of distances and distance numbers of the vertices of a graph. The notion of ”distance tree of a vertex” is briefly reviewed.
By means of this formalism the change in the distance and Wiener numbers upon some elementary graph operations (covering several graphs upon a common induced subgraph, joining graphs by edges, and subdivision of an edge by a vertex) are studied. Various general formulae for distance and Wiener numbers of the graphs resulting after these operations are given. They provide deep and new insights into the modelling of distance and Wiener numbers and their interplay.