A cycle on \(n\) ݊vertices is isomorphic to its complement. The value of…

Answer: 5

Reason:

• The cycle on n vertices is 2-regular, so every vertex has degree 2.

• The complement of a k-regular graph on n vertices is (n-1-k)-regular. Therefore, the complement of the cycle is (n-3)-regular.

• If the cycle is isomorphic to its complement, their vertex degrees must match, so 2 = n - 3. Solving gives n = 5.

• Check: the complement of the 5-cycle is itself a 5-cycle, so n = 5 indeed works.