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.
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