rotnoder

rotnoder is a term used in graph theory and network science to describe nodes that participate in rotational symmetries of a structure. The term combines rotation and node, and is used to refer to the members of orbits formed under the action of a rotation group on the graph. In this context, a rotnoder is a vertex that maps to other vertices within the same symmetry orbit when a rotation from the group is applied; orbits of size one correspond to fixed points under the rotation.

The concept is used to analyze symmetry within graphs and geometric embeddings. Identifying rotnoder orbits usually involves determining the automorphism group of the graph or the symmetry group of its embedding, and examining the cycle structure of the induced rotations. Algorithms from group theory and graph isomorphism are commonly employed for this task.

rotnoder can aid in reducing computational complexity by exploiting symmetry, in network design with desired symmetric properties, and in the study of physical systems where rotational invariance is important, such as crystallography and molecular modeling. In practice, the term is not universally standardized; some authors use alternative terminology such as symmetry or orbit representatives to describe the same ideas. Therefore, whenever rotnoder is used, it is common to define the rotation group and the corresponding orbits explicitly.

See also: automorphism, symmetry, group action, orbit, graph embedding.