If $G$ is a finite, connected simple graph then is there an expression for the average geodesic length?


That is suppose I know two nodes $n\_1$ and $n\_2$, the number of edges in my graph and at those points and the number of vertices then is there a formula giving reasonable bounds on the geodesic connecting $n\_1$ to $n\_2's$ length?



Asymptotic formula for average geodesic length on graph?
There is no formula depending just on the number of vertices (or even the number of vertices plus number of edges), and for a $d$-regular graph, for example, the average could range anywhere from linear in the number of vertices to logarithmic - this has to do with the expansion properties of the graph. 


