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