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 most2 3 α , if n ≤ 2 α − 1 , and at most α + 2 , if n > 2 α − 1 . 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.