On mean distance and girth
We bound the mean distance in a connected graph which is not a tree in terms of its order n n and its girth g g . On one hand, we show that the mean distance is at most n + 1 3 ? g ( g 2 ? 4 ) 12 n ( n ? 1 ) ? g ( g ? 2 ) ( n ? g ) 2 n ( n ? 1 ) if g g is even and at most n + 1 3 ? g ( g 2 ? 1 ) 12 n ( n ? 1 ) ? ( g ? 1 ) 2 ( n ? g ) 2 n ( n ? 1 ) if g g is odd. On the other hand, we prove that the mean distance is at least n g 4 ( n ? 1 ) unless G G is an odd cycle. This resolves two conjectures of AutoGraphiX.
mean distance girth