Nordhaus−Gaddum type results for the Harary index of graphs

The Harary index H(G) of a connected graph Gis defined as H(G) = EU,VEV(G) , where d(u, v) is the distance between vertices u and v of G. The Steiner distance in agraph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph G of order at least 2 and S SV(G), the Steiner distance do(s) of the vertices of S is the minimum size of a connected subgraph whose vertex set contains S. Recently, Furtula, Gutman, and Katanić introduced the concept of Steiner Harary index and gave its chemical applications. The kcenter Steiner Harary index SH (G) of G is defined by SH (G) = sv(G),[S1=k doces. In this paper, we get the sharp upper and lower bounds for SH (G) +SH (G) and SH (G).SH:(G), valid for any connected graph G whose complement G is also connected.