Density results for Graovac-Pisanski’s distance number - Abrams, Lowell (Ars mathematica contemporanea, 2021)
The sum of distances between every pair of vertices in a graph ?$G$? is called the Wiener index of ?$G$?. This graph invariant was initially utilized to predict certain physico-chemical properties of organic compounds. However, the Wiener index of ?$G$? does not account for any of its symmetries, which are also known to effect these physico-chemical properties. A. Graovac and I. Pisanski On the Wiener index of a graph, J. Math. Chem. 8, No. 1, 53-62 (1991; doi:10.1007/BF01166923) modified the Wiener index of ?$G$? to measure the average distance each vertex is displaced under the elements of the symmetry group of ?$G$?; we call this the Graovac-Pisanski (GP) distance number of ?$G$?. In this article, we prove that the set of all GP distance numbers of graphs with isomorphic symmetry groups is dense in a half-line. Moreover, for each finite group ?$Γ$? and each rational number ?$q$? within this half-line, we present a construction for a graph whose GP distance number is ?$q$? and whose symmetry group is isomorphic to ?$Γ$?. This construction results in graphs whose vertex orbits are not connected; we also consider an analogous construction which ensures that all vertex orbits are connected.
Keywords: chemical graph theory; distance number; Graovac-Pisanski index; graph automorphism group; grupa avtomorfizmov grafa; indeks Graovac-Pisanskega; kemijska teorija grafov; razdaljno število; Wiener index; Wienerjev indeks