DYAD ALGEBRA AND MULTIPLICATION OF GRAPHS .1. DIRECTED-GRAPHS

The ket-bra algebra for quantum mechanics and for the quantum chemistry in valence shells was made by this author fully covariant recently. The resulting ''principle of linear cavariance'' allowed diverse approaches such as molecular orbital, valence bond, localized orbital theories to come out as special cases in particular basis frames not necessarily orthonormal. The principal also led to the pictorial VIF (valency interaction formula) methods for deducing qualitative quantum chemistry directly from the structural formulas of molecules. The present set of two papers (II on undirected graphs) develops graphs and graph rules for abstract linear vector spaces, bras, kets, and abstract operators as ket-bra dyads. Multiplications of such operators are carried out with graphs of two kinds of lines and two kinds of vertices. The theorems are demonstrated on some examples and are useful, e.g., with the recent method of moments and in deriving Lie algebras pertinent to quantum chemistry.