Answer to Question #306181 in Discrete Mathematics for chuppu

Question

Prove or disprove that there exists exactly two regular graphs

Answer 1

For a graph to be 2-regular, it has to have at least 3 vertices. Considering that, if we take 3 vertices from the 5 vertices available and make them 2-regular, we are left with two other vertices that cannot be 2-regular by themselves:

Similarly, if we take 4 vertices and make them 2-regular, we are left with 1 vertex that cannot be 2-regular by itself:

Hence the only way of getting a 2-regular graph on 5 vertices is to join all 5 of them in a cycle.