Number of Simple Graph with N Vertices and M Edges

Given two integers N and M, the task is to count the number of simple undirected graphs that can be drawn with N vertices and M edges. A simple graph is a graph that does not contain multiple edges and self-loops.

Examples: 

Input: N = 3, M = 1 
Output: 3 
The 3 graphs are {1-2, 3}, {2-3, 1}, {1-3, 2}.

Input: N = 5, M = 1 
Output: 10 

Approach: The N vertices are numbered from 1 to N. As there are no self-loops or multiple edges, the edge must be present between two different vertices. So the number of ways we can choose two different vertices is ^NC₂ which is equal to (N * (N - 1)) / 2. Assume it P. 

Now M edges must be used with these pairs of vertices, so the number of ways to choose M pairs of vertices between P pairs will be ^PC_M. 

If P < M then the answer will be 0 as the extra edges can not be left alone.

Below is the implementation of the above approach: 

10

Complexity Analysis:

• Time Complexity: O(M), where M is the number of edges.
• Space Complexity: O(1), since no extra space has been taken.