A graph is self-complementary if it is isomorphic to its complement. For all…

Key idea: a self-complementary graph must have half of all possible edges, so the number of edges must be an integer equal to n(n-1)/4.

• Step 1: Let total possible edges be C(n,2) = n(n−1)/2. If a graph is isomorphic to its complement it must have half of those edges, so required edges m = n(n−1)/4.

• Step 2: Therefore n(n−1) must be divisible by 4. Check n modulo 4:

  • If n ≡ 0 (mod 4) then n(n−1) ≡ 0, so divisible by 4.

  • If n ≡ 1 (mod 4) then n−1 ≡ 0, so n(n−1) ≡ 0, divisible by 4.

  • If n ≡ 2 (mod 4) or n ≡ 3 (mod 4) then n(n−1) ≡ 2 (mod 4), not divisible by 4.

• Conclusion: n must be congruent to 0 or 1 modulo 4.

Examples: n = 1, 4, 5, 8, ... all satisfy the condition.