how many non isomorphic directed simple graphs are there wit

how many non isomorphic directed simple graphs are there with \'n\' vertices, when n=2, n=3 and n=4

Solution

When number of vertices are 2, there are 2 non isomorphic directed simple graphs. For any graph on with two vertices has either one edge or zero edges. Any pair of graphs on two vertices with zero edges are isomorphic: send a point to a point, and the other point to the other point. Same is true for one edge. These two cases are not isomorphic since they don’t have the same number of edges.

When number of vertices are 3, there are 4 non isomorphic direct simple graphs. Since there is exactly one graph on three vertices for any number of edges between 0 and 3, giving three isomorphism classes. Hence 4 non-ishomorphic graphs.

When number of vertices are 4, there are 10 non-isomorphic directed graphs. We get 1 graph for each 0, 1, 5 and 6 edges and we get 2 graphs for each 2, 3 and 4 edges. Hence total number of non-isomorphic graphs is 10.