  We present a simple observation showing that the heat kernel on a locally
finite graph behaves for short times $t$ roughly like $t^d$, where $d$ is the
combinatorial distance. This is very different from the classical Varadhan type
behavior on manifolds. Moreover, this also gives that short time behavior and
global behavior of the heat kernel are governed by two different metrics
whenever the degree of the graph is not uniformly bounded.
