On the Expressive Power of Homomorphism Counts

TL;DR

This paper explores the expressive differences between left and right homomorphism profiles of graphs, revealing that certain graph equivalences are captured by one profile but not the other, thus deepening understanding of graph isomorphism relaxations.

Contribution

It demonstrates that restrictions of the right profile cannot capture fractional isomorphism, counting logic equivalences, or co-spectrality, unlike the left profile, and vice versa for chromatic equivalence.

Findings

Right profile restrictions do not capture fractional isomorphism.

Right profile restrictions do not capture counting logic equivalences.

Left profile restrictions do not capture chromatic equivalence.

Abstract

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