On Mean Distance and Girth

Abstract: We bound the mean distance in a connected graph which is not a tree in function of its order $n$ and its girth $g$. On one hand, we show that mean distance is at most $\frac{n+1}{3}-\frac{g(g^2-4)}{12n(n-1)}$ if $g$ is even and at most $\frac{n+1}{3}-\frac{g(g^2-1)}{12n(n-1)}$ if $g$ is odd. On the other hand, we prove that mean distance is at least $\frac{ng}{4(n-1)}$ unless $G$ is an odd cycle.