A classical result by Lovász 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. Dvorák 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.

In the past few years, Internet of Things (IoT) has emerged as an outstanding technology and can be considered as the backbone of the smart world era as it connects physical devices with the internet. On the other hand there are these intelligent conversation interfaces that use a dialogue system to have a conversation with the users and are already used widely. In this paper, we focus on integration of chatbots and IoT to address a critical problem like air quality awareness. We present the architecture of the chatbot and its implementation on an instant messaging application. The chatbot not only provides air quality, temperature and humidity information to the users but also provides services like subscription to air quality monitoring nodes in a particular area or any part of the service region, alarm services, threshold settings, geoquery and recommendation based on the pollutant levels. A detailed explanation of the scenario of use, challenges identified during development process and possible future directions of this integration have also been addressed.