Connection Matrices for MSOL-Definable Structural Invariants

Connection matrices of graph parameters were first introduced by M. Freedman, L. Lovász and A. Schrijver (2007) to study the question which graph parameters can be represented as counting functions of weighted homomorphisms. The rows and columns of a connection matrix M(f, ) of a graph parameter f and a binary operation are indexed by all finite (labeled) graphs Gi and the entry at (Gi, Gj) is given by the value of f(Gi Gj). Connection matrices turned out to be a very powerful tool for studying graph parameters in general. B. Godlin, T. Kotek and J.A. Makowsky (2008) noticed that connection matrices can be defined for general relational structures and binary operations between them, and for general structural parameters. They proved that for structural parameters f definable in Monadic Second Order Logic, (MSOL) and binary operations compatible with MSOL, the connection matrix M(f, ) has always finite rank. In this talk we discuss several applications of this Finite Rank Theorem, and outline ideas for further research.