A Combinatorial Approach to Study the Nordhaus–Guddum-Type Results for Steiner Degree Distance

In 1994, Dobrynin and Kochetova introduced the concept of degree distance DD(Γ) a connected graph Γ. Let dΓ(S) be Steiner k-distance S⊆V(Γ). The Wiener k-index or k-center indexSWk(Γ) Γ is defined by SWk(Γ)=∑|S|=kS⊆V(Γ)dΓ(S). distanceSDDk(Γ) SDDk(Γ)=∑|S|=kS⊆V(Γ)∑v∈SdegΓ(v)dΓ(S), where degΓ(v) vertex v in this paper, we consider Nordhaus–Gaddum-type results for SWk(Γ) SDDk(Γ). Upper bounds on SWk(Γ)+SWk(Γ¯) SWk(Γ)·SWk(Γ¯) are obtained compared with previous bounds. We present sharp upper lower SDDk(Γ)+SDDk(Γ¯) SDDk(Γ)·SDDk(Γ¯) order n maximum Δ minimum δ. Some classes attaining these also given.