## Abstract

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.