On the Expressive Power of Homomorphism Counts

A classical result by Lov\'asz asserts that two graphs $G$ and $H$ are isomorphic if and only if they have the same left-homomorphism vector, that is, for every graph $F$, the number of homomorphisms from $F$ to $G$ coincides with the number of homomorphisms from $F$ to $H$. Dell, Grohe, and Rattan showed that restrictions of the left-homomorphism vector to a class of graphs can capture several different relaxations of isomorphism, including co-spectrality (i.e., two graphs having the same characteristic polynomial), fractional isomorphism and, more broadly, equivalence in counting logics with a fixed number of variables. On the other side, a result by Chaudhuri and Vardi asserts that isomorphism is also captured by the right-homomorphism vector, that is, two graphs $G$ and $H$ are isomorphic if and only if for every graph $F$, the number of homomorphisms from $G$ to $F$ coincides with the number of homomorphisms from $H$ to $F$. In this paper, we embark on a study of the restrictions of the right-homomorphism vector by investigating relaxations of isomorphism that can or cannot be captured by restricting the right-homomorphism vector to a fixed class of graphs. Our results unveil striking differences between the expressive power of the left-homomorphism vector and the right-homomorphism vector. We show that co-spectrality, fractional isomorphism, and equivalence in counting logics with a fixed number of variables cannot be captured by restricting the right-homomorphism vector to a class of graphs. In the opposite direction, we show that chromatic equivalence cannot be captured by restricting the left-homomorphism vector to a class of graphs, while, clearly, it can be captured by restricting the right-homomorphism vector to the class of all cliques.