The k-Szeged index of graphs

Based on (revised) Szeged index of a graph and the Steiner k-Wiener index of a tree, we introduce the k-Szeged index Sz(k)(G) and revised k-Szeged index Sz(k)(*)(G) of a graph G =(V, E), defined as Sz(k)(*)(G) = Sigma(e=uv epsilon E(G)) Sigma(k-1)(i=1)(n(u)(e)+n(0)(e)/2 i) (n(v)(e)+n(0)(e)/2 k-i) and SZ(k) (G) = Sigma(e=uv epsilon E(G)) Sigma(k-1)(i=1) (n(u)(e) i) (n(u)(e) k-i) , where n(u)(e), n(v)(e) and n(0)(e) denote respectively the number of vertices of Glying closer to vertex uthan to vertex v, the number of vertices of Glying closer to vertex vthan to vertex uand the number of vertices with equal distance to uand v. In this paper, we first determine upper and lower bounds of (revised) k-Szeged indices of a connected graph Gin terms of the numbers of vertices, edges and pendant edges, and give Nordhaus-Gaddum-type results of (revised) k-Szeged indices. Then we determine the extremal value of these indices and the corresponding extremal graphs among all complete bipartite graphs with nvertices. Finally, we discuss the upper and lower bounds of these indices for unicyclic graphs.