The Steiner diameter of a graph with prescribed girth

Let G be a connected graph of order p , and let S be a nonempty set of vertices of G . Then the Steiner distance d ( S ) of S is the minimum size of a connected subgraph of G whose vertex set contains S . If n is an integer, 2 ≤ n ≤ p , the Steiner n -diameter, d i a m n ( G ) , of G is the maximum Steiner distance of any n -subset of vertices of G . We give upper bounds on the Steiner n -diameter of G in terms of order, minimum degree δ , and girth g , of G . Moreover, we construct graphs to show that, if for given δ and g there exists a Moore graph of minimum degree δ and girth g , then the bounds are asymptotically sharp.