On distance-based topological indices used in architectural research

Abstract

Distance-based topological indices have been used in studies of molecular graphs ever since Harry Wiener introduced his now famous index back in the 1940s, with tens of such indices studied actively in current mathematical chemistry literature. Interestingly, two further distance-based invariants, the difference factor and the intelligibility, have been used in parallel in studies of graphs associated to building and urban plans since the 1970s as well. These invariants are defined in terms of integration values that represent normalized values of the sums of distances from a given vertex to all other vertices in a graph. The difference factor is defined as an entropic measure that quantifies the diversity of the sequence of integration values, while the intelligibility is defined as the Pearson correlation coefficient between sequences of vertex degrees and integration values thus quantifying the extent to which integration values, for which one has to know the structure of the whole graph, can be predicted from vertex degrees, for which one has to know only how many neighbors a vertex has. We perform here a number of computational studies of the difference factor and the intelligibility that reveal to what extent these invariants can be used as topological indices in mathematical chemistry as well.