  Let $G$ be a simple, finite graph and let $p_t(x,y)$ denote the heat kernel
on $G$. The purpose of this short note is to show that for $t \rightarrow 0^+$
$$ p_t(x,y) = \# \left\{\mbox{paths of
length}~d(x,y)~\mbox{between}~x~\mbox{and}~y\right\} \frac{t^{d(x,y)}}{d(x,y)!}
+ \mathcal{O}(t^{d(x,y)+1}),$$ where $d(x,y)$ is the usual Graph distance. This
is the discrete analogue of the classical Varadhan asymptotic for the heat
kernel on manifolds and refines a result of Keller, Lenz, M\"unch, Schmidt and
Telcs. The asymptotic behavior encapsulates additional geometric information:
if the Graph is bipartite, then the next term in the expansion is negative.
