Bond valence and finite graph symmetries

Abstract

Abstract The graph theory prediction of bond valence distributions in a crystal network is reformulated in a form more familiar to chemists, that of symmetry-adapted functions. The graph theory terms relevant to this exposition are explained, including graph matrices, graph vector spaces and graph automorphism groups. It was found that the vertex and edge sets of a chemical graph form representations of the graph automorphism group. These representations may be expressed as a sum of irreducible representations using the group character table in the usual way. Alternatively the classes of the individual orbits spanned may be used to derive these irreducible representations. The orthogonal cut and cycle subspaces of the edge space can be defined so as to also transform as a sum of irreducible representations. Hence appropriate symmetry-adapted orthogonal basis vectors may be chosen for the vertex and edge spaces. Removing the usual Equal Valence Rule constraint creates a solution space, a range of possible solutions for a particular prob lem. This solution space may be found by combining a set of contributions from the cut space vectors fixed by the atom valences, with a convex polytope in the cycle space which is defined by the non-negativity of the individual bond valences. The complete procedure is illustrated for the β-Ga2O3 structure.