The Wiener number of powers of the Mycielskian

The Wiener number of a graph G is defined as ${\displaystyle \frac{1}{2}{\sum }_{u,v\in V\left(G\right)}d(u,v)}$, d the distance function on G. The Wiener number has important applications in chemistry. We determine a formula for the Wiener number of an important graph family, namely, the Mycielskians μ(G) of graphs G. Using this, we show that for k ≥ 1, ${\displaystyle W(\mu \left(S{ₙ}^k\right))\le W(\mu \left(T{ₙ}^k\right))\le W(\mu \left(P{ₙ}^k\right))}$, where Sₙ, Tₙ and Pₙ denote a star, a general tree and a path on n vertices respectively. We also obtain Nordhaus-Gaddum type inequality for the Wiener number of ${\displaystyle \mu \left(G^k\right)}$.