### Density results for Graovac-Pisanski’s distance number

#### Abstract

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. Graovac and Pisanski 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.

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DOI: https://doi.org/10.26493/1855-3974.2351.07b

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications