On the Expressive Power of Homomorphism Counts
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A classical result by Lovász 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.
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