Let $G$ be a graph and $v$ one of its vertices.

Are there any known formulas or fast algorithms for calculating the expected length of a random path in $G$ starting in $v$?

In each step we choose a neighbour of the current vertex uniformly, if it has already been visited we stop the process, otherwise we move to that vertex.

For example, if our graph is a path with $3$ vertices and $v$ is one of the edges, the expected length is $\frac{3+2}{2}$

I am also interested in formulas for the expected length if the initial vertex is chosen uniformly. Thank you kindly.