  A bipartite graph $H$ is said to have Sidorenko's property if the probability
that the uniform random mapping from $V(H)$ to the vertex set of any graph $G$
is a homomorphism is at least the product over all edges in $H$ of the
probability that the edge is mapped to an edge of $G$. In this paper, we
provide three distinct families of bipartite graphs that have Sidorenko's
property. First, using branching random walks, we develop an embedding
algorithm which allows us to prove that bipartite graphs admitting a certain
type of tree decomposition have Sidorenko's property. Second, we use the
concept of locally dense graphs to prove that subdivisions of certain graphs,
including cliques, have Sidorenko's property. Third, we prove that if $H$ has
Sidorenko's property, then the Cartesian product of $H$ with an even cycle also
has Sidorenko's property.
