Counting Walks and Graph Homomorphisms via Markov Chains and Importance Sampling

Hoffman [7] proved a matrix inequality that yields a useful upper bound on the number of walks in a graph. Sidorenko [14] extended the bound on the number of walks to a bound on the number of homomorphisms from a tree to a graph. In this expository note, we give a short probabilistic proof of both results, using the basic identity of importance sampling.