This theorem states that Gamma be a finite connected simple graph, and let C Gamma be the symmetric matrix indexed by the vertices of Gamma and defined by (C ) vv &= 2,\\ (C ) vw &= -1 v w ,\\ (C ) vw &= 0 v w v,w .. It is used in the structure and classification of finite-dimensional Lie algebras, especially in arguments involving Cartan data, root systems, Weyl groups, and Dynkin diagrams.