Dirac showed that in a $(k-1)$-connected graph there is a path through all the k vertices. The k-path-connectivity ${\pi }_k(G)$ of a graph G, which is a generalization of Dirac's notion, was introduced by Hager in 1986. Denote by $G\circ H$ the lexicographic product of two graphs G and H. In this paper, we prove that ${\pi }_3(G\circ H)\ge {\pi }_3(G)\lfloor (|V(H)|-1)/2\rfloor +1$ for any two connected graphs G and H. Moreover, the bound is sharp. We also derive an upper bound of ${\pi }_3(G\circ H)$, that is, ${\pi }_3(G\circ H)\le 2{\pi }_3(G)|V(H)|$.

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