For instance, an infinite path graph with one endpoint has only a single odd-degree vertex rather than having an even number of such vertices. However, it is possible to formulate a version of the handshaking lemma using the concept of an end, an equivalence class of semi-infinite paths ("rays") considering two rays as equivalent when there exists a third ray that uses infinitely many vertices from each of them.
For example, an infinite path graph with one endpoint has only one vertex with an odd degree, rather than having an even number. However, it is possible to formulate a version of the handshaking lemma using the concept of an end, where two semi-infinite paths ("rays") are considered equivalent if there exists another ray that uses infinitely many vertices from each.