Let $G, H$ be simple, undirected graphs without loops. We say that $G, H$ have the same homomorphism fingerprint if $|\text{Hom}(X, G)| = |\text{Hom}(X, H)|$ for all graphs $X$. (By graph homomorphisms I mean edge preserving maps; so for instance $\text{Hom}(X,K\_2)=\emptyset$ if $\chi(X) > 2$).


From Lovasz [1] we have the following theorem: If $G, H$ are finite and they have the same homomorphism fingerprint, then $G\cong H$.


Are there non-isomorphic infinite graphs that have the same homomorphism fingerprint?


[1] L. Lovasz, Operations with structures,Acta Math. Hungar. 18(1967),
321–328



"Homomorphism fingerprint" for graphs
In any category, if $G$ and $H$ are objects such that there exist monic maps $G\to H$ and $H\to G$, then $|\text{Hom}(X, G)| = |\text{Hom}(X, H)|$ for all $X$. There are plenty of pairs of non-isomorphic infinite graphs with this property.


