Clique with one vertex

Question

Is there a clique possible in Graph with one vertex? I mean, is a singleton vertex in itself is a complete sub-graph and can be called a clique.

Comments

depends on graph

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As in?

Answer 1

Definition of Clique : Clique is  a subset of the set of vertices in an undirected graph such that any two distinct vertices in the clique are adjacent.

Now come to the definition of Vacuous truth.

In mathematics and logic, a vacuous truth is a statement that asserts that all members of an empty set have a certain property. For example, the statement "all cell phones in the room are turned off" will be true whenever there are no cell phones in the room. In this case, the statement "all cell phones in the room are turned on" would also be vacuously true, as would the conjunction of the two: "all cell phones in the room are turned on and turned off". 

[ Taken from https://en.wikipedia.org/wiki/Vacuous_truth ]

Now, the statement a subset V₁ of V(G) is a clique is true if  all distinct pairs of vertices in V1 are adjacent to each other.

So, the statement V₁ is a clique is vacuously true when there are no pairs in V₁ .

So, a subset having only one vertex is a clique is vacuously true because there are no pairs in that vertex.

So, a singleton vertex can be called a clique. [ It is vacuously true.]

Note that a singleton vertex can be called an independent set. [ It is also vacuously true ]

Comments

@kushagra , the statement "all cell phones in the room are turned off" will be true whenever there are no cell phones in the room. In this case, the statement "all cell phones in the room are turned on" would also be vacuously true

based on this fact , can't we say that  "the statement V₁ is not a clique is also vacuously true when there are no pairs in V₁" ?

can u please tell me that for edgeless graphs or null graphs , clique(s) exist or not ?

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 In your comment you have written

the statement "all cell phones in the room are turned off" will be true whenever there are no cell phones in the room. In this case, the statement "all cell phones in the room are turned on" would also be vacuously true

based on this fact , can't we say that  "the statement V₁ is not a clique is also vacuously true when there are no pairs in V₁" ?

No, you can't say that if you want to say something based on the above fact then you have to go to the definition of clique.

S1 : V₁ is a clique.

We can write S1 as 

S1 : V₁ is a subgraph of G such that all pair of vertices in V₁ are adjacent to each other. 

Now, if you want to frame a sentence similar to " all cell phones are turned on "

That will be

S₂ : V₁ is a subgraph of G such that all pair of vertices in V₁ are not adjacent to each other.

That is 

S₂ : V₁ is an independent set.

S₂ can be vacuously true. (which I have mentioned in my answer.)

And since singleton vertex is a clique so edgeless graphs contain cliques.

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got it...Thank you !