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The complementary graph G'  of a simple graph G has the same vertices as G. Two vertices are adjacent in G'  if and only if they are not adjacent in G. Define Qn' (Hypercube complement).

Answer given :-The graph whose vertices are bit strings of length n and two vertices are adjacent if the bit string represented by them differe by more than one bit.

I want to understand that whether the complement graph will have self loops?Because the answer given doesn't consider self loops.I mean why are we not considering the bit strings that are differing by 0 bit,as these are also not there in original graph ,so it must be in complementary graph?
asked in Graph Theory by Loyal (4.9k points)   | 46 views

The definition itself says Simple graph it can't have self loops

Simple graph: No self loop & No Multiedges between 2 vertices

Multi graph: Can have multieedges between 2 vertices but no self loop

General graph or Pseudograph: Can have both

But it was not mentioned that complement graph of simple graph has to be simple.
They are talking about 2 adjacent vertices,if there was and edge in G then it can't be there in G' .So if we take 2 vertices v1 and v2 ,if there were an edge between them in G it shouldn't be there in G'

Moreover when you do G U G' you should get complete graph. Even complete graph can't have self loop
Got it .thanks:)

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