Nordhaus-Guddum type results for the Steiner Gutman index of graphs

TL;DR

This paper establishes new bounds and Nordhaus-Gaddum-type results for the Steiner Gutman index of graphs, extending the understanding of this graph invariant and its behavior under graph complementation.

Contribution

It introduces sharp bounds for the Steiner Gutman index and explores Nordhaus-Gaddum results, advancing the theoretical understanding of this graph parameter.

Findings

Derived sharp bounds for SGut_k(G)

Established Nordhaus-Gaddum inequalities for SGut_k

Analyzed the index for graphs with given order, edges, and degrees

Abstract

Building upon the notion of Gutman index $\operatorname{SGut}(G)$, Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph $G$. The \emph{Steiner Gutman $k$-index} $\operatorname{SGut}_k(G)$ of $G$ is defined by $\operatorname{SGut}_k(G)$ $=\sum_{S\subseteq V(G), \ |S|=k}\left(\prod_{v\in S}deg_G(v)\right) d_G(S)$, in which $d_G(S)$ is the Steiner distance of $S$ and $deg_G(v)$ is the degree of $v$ in $G$. In this paper, we derive new sharp upper and lower bounds on $\operatorname{SGut}_k$, and then investigate the Nordhaus-Gaddum-type results for the parameter $\operatorname{SGut}_k$. We obtain sharp upper and lower bounds of $\operatorname{SGut}_k(G)+\operatorname{SGut}_k(\overline{G})$ and $\operatorname{SGut}_k(G)\cdot \operatorname{SGut}_k(\overline{G})$ for a connected graph $G$ of order $n$, $m$ edges and maximum degree $\Delta$, minimum degree $\delta$.