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

by Hongfang Liu, Jinxia Liang, Yuhu Liu and Kinkar Chandra Das

Mathematics 2023, 11(3), 738; https://doi.org/10.3390/math11030738 - 1 Feb 2023

Cited by 1 | Viewed by 1665

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 \subseteq V (Γ)

. The Steiner [...] Read more.

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 \subseteq V (Γ)

. The Steiner Wiener k-index or k-center Steiner Wiener index

{SW}_{k} (Γ)

Γ

is defined by

{SW}_{k} (Γ) = \sum_{\overset{S \subseteq V (Γ)}{| S | = k}} d_{Γ} (S) .

The k-center Steiner degree distance

{SDD}_{k} (Γ)

of a connected graph

Γ

is defined by

{SDD}_{k} (Γ) = \sum_{\overset{S \subseteq V (Γ)}{| S | = k}} [\sum_{v \in 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} (\bar{Γ})

and

{SW}_{k} (Γ) \cdot {SW}_{k} (\bar{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} (\bar{Γ})

and

{SDD}_{k} (Γ) \cdot {SDD}_{k} (\bar{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. Full article

(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory, 2nd Edition)

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