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

Hongfang Liu, Jinxia Liang, Yuhu Liu and Kinkar Chandra Das ()
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Hongfang Liu: School of Education, Qinghai Normal University, Xining 810008, China
Jinxia Liang: School of Mathematics and Statistic, Qinghai Normal University, Xining 810008, China
Yuhu Liu: School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China
Kinkar Chandra Das: Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea

Mathematics, 2023, vol. 11, issue 3, 1-19

Abstract: 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.

Keywords: distance; Steiner distance; degree distance; Steiner Wiener k -index; k -center Steiner degree distance; Nordhaus–Gaddum-type result (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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