Abstract
Let G be a finite, connected graph and v a vertex of G. The average distance and the eccentricity of v in G are defined as the arithmetic mean and the maximum, respectively, of the distances from v to all other vertices of G. The proximity of G and the radius of G are defined as the minimum of the average distances and the eccentricities over all vertices of G.
In this paper, we establish an upper bound on the proximity of a 2-connected outerplanar graphs in terms of order and maximum face length. This bound is sharp apart from a small additive constant.
It is known that the radius of a maximal outerplanar graph is at most ⌊n4⌋+1. In the second part of this paper we show that this bound on the radius holds for a much larger subclass of outerplanar graphs, for all 2-connected outerplanar graphs of order n whose maximum face length does not exceed n+24.