  In this paper, we study the graph classification problem from the graph
homomorphism perspective. We consider the homomorphisms from $F$ to $G$, where
$G$ is a graph of interest (e.g. molecules or social networks) and $F$ belongs
to some family of graphs (e.g. paths or non-isomorphic trees). We show that
graph homomorphism numbers provide a natural invariant (isomorphism invariant
and $\mathcal{F}$-invariant) embedding maps which can be used for graph
classification. Viewing the expressive power of a graph classifier by the
$\mathcal{F}$-indistinguishable concept, we prove the universality property of
graph homomorphism vectors in approximating $\mathcal{F}$-invariant functions.
In practice, by choosing $\mathcal{F}$ whose elements have bounded tree-width,
we show that the homomorphism method is efficient compared with other methods.
