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.
To demonstrate, consider an infinite path graph with a single endpoint. This results in only one vertex having an odd degree, rather than an even number. Nevertheless, it is feasible to develop a version of the handshaking lemma based on the concept of an end, where two semi-infinite paths ("rays") are considered equivalent if there exists another ray that utilizes infinitely many vertices from each.