  We provide an analysis of the expected meeting time of two independent random
walks on a regular graph. For 1-D circle and 2-D torus graphs, we show that the
expected meeting time can be expressed as the sum of the inverse of non-zero
eigenvalues of a suitably defined Laplacian matrix. We also conjecture based on
empirical evidence that this result holds more generally for simple random
walks on arbitrary regular graphs. Further, we show that the expected meeting
time for the 1-D circle of size $N$ is $\Theta(N^2)$, and for a 2-D $N \times
N$ torus it is $\Theta(N^2 log N)$.
