GATE CS 2015 Set 2 — Question 61
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Engineering Mathematics → Graph Theory (Math) → Graph Terminology (Degree, Paths, Cycles)
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Question
A graph is self-complementary if it is isomorphic to its complement. For all self-complementary graphs on vertices, is
Correct answer
(D) Congruent to 0 ±od 4, or, 1 ±od 4.
Solution
A graph is self-complementary if . This implies that the number of edges in is equal to the number of edges in .
Since (total edges in ), we have:
For the number of edges to be an integer, must be divisible by 4. Since and are consecutive integers, one is odd and the other is even. The odd number cannot contribute to the factor 4, so the even number must be divisible by 4.
Case 1: is divisible by 4 .
Case 2: is divisible by 4 .
Since (total edges in ), we have:
For the number of edges to be an integer, must be divisible by 4. Since and are consecutive integers, one is odd and the other is even. The odd number cannot contribute to the factor 4, so the even number must be divisible by 4.
Case 1: is divisible by 4 .
Case 2: is divisible by 4 .
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