Counting homomorphisms between graphs (often with weights) comes up in a wide variety of areas, including extremal graph theory, properties of graph products, partition functions in statistical physics and property testing of large graphs.
In this paper we survey recent developments in the study of homomorphism numbers, including the characterization of the homomorphism numbers in terms of the semidefiniteness of “connection matrices”, and some applications of this fact in extremal graph theory.
We define a distance of two graphs in terms of similarity of their global structure, which also reflects the closeness of (appropriately scaled) homomorphism numbers into the two graphs. We use homomorphism numbers to define convergence of a sequence of graphs, and show that a graph sequence is convergent if and only if it is Cauchy in this distance. Every convergent graph sequence has a limit in the form of a symmetric measurable function in two variables. We use these notions of distance and graph limits to give a general theory for parameter testing.
The convergence can also be characterized in terms of mappings of the graphs into fixed small graphs, which is strongly connected to important parameters like ground state energy in statistical physics, and to weighted maximum cut problems in computer science.