The upper bounds on the Steiner k-Wiener index in terms of minimum and maximum degrees

For $$k \in {\mathbb {N}},$$ k ∈ N , Ali et al. (Discrete Appl Math 160:1845-1850, 2012) introduce the Steiner k-Wiener index $$SW_{k}(G)=\sum _{S\in V(G)} d(S),$$ S W k ( G ) = ∑ S ∈ V ( G ) d ( S ) , where d(S) is the minimum size of a connected subgraph of G containing the vertices of S. The average Steiner k-distance $$\mu _{k}(G)$$ μ k ( G ) of G is defined as $$\genfrac(){0.0pt}1{n}{k}^{-1} SW_{k}(G)$$ n k - 1 S W k ( G ) . In this paper, we give some upper bounds on $$SW_{k}(G)$$ S W k ( G ) and $$\mu _{k}(G)$$ μ k ( G ) in terms of minimum degree, maximum degree and girth in a triangle-free or a $$C_{4}$$ C 4 -free graph.