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Connection Matrices and the Definability of Graph Parameters Kotek, Tomer Makowsky, Johann A. Model theory finite model theory graph invariants In this paper we extend the Finite Rank Theorem for connection matrices of graph parameters definable in Monadic Second Order Logic with modular counting CMSOL of B. Godlin, T. Kotek and J.A. Makowsky (2008 and 2009), and demonstrate its vast applicability in simplifying known and new non-definability results of graph properties and finding new non-definability results for graph parameters. We also prove a Feferman-Vaught Theorem for the logic CFOL, First Order Logic with the modular counting quantifiers. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Tomer Kotek and Johann A. Makowsky 2012 Is Part Of LIPIcs, Volume 16, Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL (2012) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CSL.2012.411 urn:nbn:de:0030-drops-36871 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2012.411 eng https://creativecommons.org/licenses/by-nc-nd/3.0/legalcode