Nordhaus–gaddum-type results for the steiner gutman index of graphs

Building upon the notion of the Gutman index SGut(G), 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 (G) of G is defined by SGut_k (G) = ∑_{S⊆V(G),|S|=k} (∏_v∈S deg_G (v)) 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 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 (G) + SGut_k (G) and SGut_k (G) · SGut_k (G) for a connected graph G of order n, m edges, maximum degree ∆ and minimum degree δ.