Minimum average distance of strong orientations of graphs

Peter Dankelmann, Ortrud R. Oellermann, Jian-Liang Wu
2004 Discrete Applied Mathematics  
The average distance of a graph (strong digraph) G, denoted by (G) is the average, among the distances between all pairs (ordered pairs) of vertices of G. If G is a 2-edge-connected graph, then˜ min (G) is the minimum average distance taken over all strong orientations of G. A lower bound for˜ min (G) in terms of the order, size, girth and average distance of G is established and shown to be sharp for several complete multipartite graphs. It is shown that there is no upper bound for˜ min (G) in
more » ... terms of (G). However, if every edge of G lies on 3-cycle, then it is shown that˜ min (G) 6 7 4 (G). This bound is improved for maximal planar graphs to 5 3 (G) and even further to 3 2 (G) for eulerian maximal planar graphs and for outerplanar graphs with the property that every edge lies on 3-cycle. In the last case the bound is shown to be sharp.
doi:10.1016/j.dam.2004.01.005 fatcat:tbhv4fceybbd7bxe6y6qexquum