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A graph is self-complementary if it is isomorphic to its complement. For all self-complementary graphs on n vertices, n is
- a)A multiple of 4
- b)Even
- c)Odd
- d)Congruent to 0 mod 4, or 1 mod 4
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
A graph is self-complementary if it is isomorphic to its complement. F...
An n-vertex self-complementary graph has exactly half number of edges of the complete graph, i.e., n(n − 1)/4 edges, and (if there is more than one vertex) it must have diameter either 2 or 3. Since n(n −1) must be divisible by 4, n must be congruent to 0 or 1 mod 4; for instance, a 6-vertex graph cannot be self-complementary.
Most Upvoted Answer
A graph is self-complementary if it is isomorphic to its complement. F...
Explanation:
In order to understand why the correct answer is option 'D', let's break down the given options and analyze them one by one.
Option A: A multiple of 4
A graph being self-complementary doesn't have any direct relation to the number of vertices being a multiple of 4. There are self-complementary graphs with any number of vertices, not just multiples of 4. Therefore, option A is incorrect.
Option B: Even
This option is also incorrect because self-complementary graphs can have an odd number of vertices. There are self-complementary graphs with 3, 5, 7, etc. vertices.
Option C: Odd
Similar to the previous option, self-complementary graphs can have an odd number of vertices. Therefore, option C is incorrect.
Option D: Congruent to 0 mod 4, or 1 mod 4
This option is correct. A graph being self-complementary is related to its number of vertices modulo 4.
Let's consider the two cases:
- If the number of vertices is congruent to 0 modulo 4 (i.e., a multiple of 4), then we can divide the vertices into two equal sets of size n/2. In the complement of the graph, each vertex in one set is adjacent to every vertex in the other set. When we swap the adjacency relationships, we get an isomorphic graph, making it self-complementary.
- If the number of vertices is congruent to 1 modulo 4, then we can divide the vertices into two sets, one with (n+1)/2 vertices and the other with (n-1)/2 vertices. In the complement of the graph, each vertex in the larger set is adjacent to every vertex in the smaller set. Again, swapping the adjacency relationships results in an isomorphic graph, making it self-complementary.
Therefore, self-complementary graphs on n vertices are congruent to 0 mod 4 or 1 mod 4.
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A graph is self-complementary if it is isomorphic to its complement. For all self-complementary graphs on n vertices, n isa)A multiple of 4b)Evenc)Oddd)Congruent to 0 mod 4, or 1 mod 4Correct answer is option 'D'. Can you explain this answer?
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