Average Distance,Independence Number,and Spanning Trees

Let G be a connected graph of order n and independence number α. We prove that G has a spanning tree with average distance at most , if , and at most , if . As a corollary, we obtain, for n sufficiently large, an asymptotically sharp upper bound on the average distance of G in terms of its independence number. This bound, apart from confirming and improving on a conjecture of Graffiti 8], is a strengthening of a theorem of Chung 1], and that of Fajtlowicz and Waller 8], on average distance and independence number of a graph.