Question

Consider the set whose elements are the graphs having vertex set {1, 2, 3, 4}, and consider the relation on that set, where two graphs are equivalent provided that they have the same number of edges. How many equivalence classes are there?

Answer 1

Answer:

7

Step-by-step explanation:

Let S be the set of all graphs having vertex set  \{1,2,3,4\}. The relation \rho is defined over S such that

the graphs G and H are equivalent provided that they have same number of edges. Then, the number of equivalence classes depends on how many edges can be there in the vertex set \{1,2,3,4\} .

The number of edges is 0 forms a disconnected graph which makes an equivalent class.

The graphs of 1 edge makes an equivalent class.

The graphs of 2 edges makes an equivalent class.

The graphs of 3 edges makes an equivalent class.

The graphs of 4 edges makes an equivalent class.

The graphs of 5 edges makes an equivalent class.

In similar way, the only graph of 6 edges is complete graph which forms another equivalent class.

Hence,the total number of equivalent classes is 7.