Cycle graph

This article defines a property that can be evaluated to true/false for any undirected graph, and is invariant under graph isomorphism. Note that the term "undirected graph" as used here does not allow for loops or parallel edges, so there can be at most one edge between two distinct vertices, the edge is completely described by the vertices it joins, and there can be no edge from a vertex to itself.
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Definition

For a finite graph with three or more vertices

An undirected graph with three or more (but finitely many) vertices is termed a cycle graph if it satisfies the following equivalent conditions:

1. It is a connected graph as well as a 2-regular graph, i.e., every vertex has degree two.
2. We can arrange a bijection between the vertex set and the group of integers modulo n (where $n$ is the number of vertices) such that there is an edge between two vertices if and only if the corresponding group elements differ by 1 modulo $n$.
3. A slightly slicker version of (2): it is the Cayley graph of a finite cyclic group of order three or more where the generating set is chosen as a cyclic generator and its inverse.

Note that, up to graph isomorphism, there is a unique cycle graph for every size of vertex set.

For the case of two or fewer vertices

For two vertices, if we allow undirected graphs and do not allow parallel edges, the correct analogue to cycle graph is the 2-clique. If we allow parallel edges, the correct analogue is a graph with two vertices and two edges between them.

For one vertex, if we allow undirected graphs and do not allow loops, the correct analogue is the only possible graph with one vertex. If we allow loops, the correct analogue is a graph with one vertex and one loop at that vertex.