In this paper we unify several existing regularity conditions for graphs, including strong regularity, k-isoregularity, and the t-vertex condition. We develop an algebraic composition/decomposition theory of regularity conditions. Using our theoretical results we show that a family of non rank 3 graphs known to satisfy the 7-vertex condition fulfills an even stronger condition, (3,7)-regularity (the notion is defined in the text). Derived from this family we obtain a new infinite family of non rank 3 strongly regular graphs satisfying the 6-vertex condition. This strengthens and generalizes previous results by Reichard.