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

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 \(\delta\).