Water clusters and some phases of ice are characterized by many isomers with similar oxygen positions, but which differ in direction of hydrogen bonds. A relationship between physical properties, like energy or magnitude of the dipole moment, and hydrogen bond arrangements has long been conjectured. The topology of the hydrogen bond network can be summarized by oriented graphs. Since scalar physical properties like the energy are invariant to symmetry operations, graphical invariants are the proper features of the hydrogen bond network which can be used to discover the correlation with physical properties. We demonstrate how graph invariants are generated and illustrate some of their formal properties. It is shown that invariants can be used to change the enumeration of symmetry-distinct hydrogen bond topologies, nominally a task whose computational cost scales like N-2, where N is the number of configurations, into an N ln N process. The utility of graph invariants is confirmed by considering two water clusters, the (H2O)(6) cage and (H2O)(20) dodecahedron, which, respectively, possess 27 and 30 026 symmetry-distinct hydrogen bond topologies associated with roughly the same oxygen atom arrangements. Physical properties of these clusters are successfully fit to a handful of graph invariants. Using a small number of isomers as a training set, the energy of other isomers of the (H2O)(20) dodecahedron can even be estimated well enough to locate phase transitions. Some preliminary results for unit cells of ice-Ih are given to illustrate the application of our results to periodic systems. (C) 2001 American Institute of Physics.