For which value of n is Wheel graph Wn regular?

In graph theory, a wheel graph Wn​ is regular only for n=3.

A graph is regular if every vertex has the same degree. In a wheel graph Wn​ (which is formed by a cycle Cn​ and one central hub vertex), the degrees are calculated as follows:

• Central Vertex (Hub): Connected to all n vertices of the cycle, so its degree is n.

• Outer Vertices (Cycle): Each is connected to two neighbors on the cycle plus the central hub, so their degree is always 3.

For the graph to be regular, the degree of the hub must match the degree of the outer vertices (n=3). When n=3, W3​becomes a 3-regular graph, which is also isomorphic to the complete graph K4​.