Wiener-type indices of Parikh word representable graphs

A new class of graphs G(w), called  Parikh word representable graphs (PWRG), corresponding to words $w$ that are finite sequence of symbols, was considered in the recent past. Several properties of these graphs have been established. In this paper, we consider these graphs corresponding to binary core words of the form $aub$ over a binary alphabet {a,b}.  We derive formulas for computing the Wiener index of the PWRG of a binary core word. Sharp bounds are established on the value of this index in terms of different parameters related to binary words over {a,b} and the corresponding PWRGs. Certain other Wiener-type indices that are variants of Wiener index are also considered. Formulas for computing these indices in the case of PWRG of a binary core word are obtained.