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

Abstract

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