A theorem on Wiener-type invariants for isometric subgraphs of hypercubes

Let d(G,k)d(G,k) be the number of pairs of vertices of a graph GG that are at distance kk, λλ a real (or complex) number, and Wλ(G)=∑k≥1d(G,k)kλ. It is proved that for a partial cube GG, Wλ+1(G)=|F|Wλ(G)−∑F∈FWλ(G∖F)Wλ+1(G)=|F|Wλ(G)−∑F∈FWλ(G∖F), where FF is the partition of E(G)E(G) induced by the Djoković–Winkler relation ΘΘ. This result extends a previously known result for trees and implies several relations for distance-based topological indices.