Problem:
Score : 500 points

Problem Statement

Rng has a connected undirected graph with N vertices.
Currently, there are M edges in the graph, and the i-th edge connects Vertices A_i and B_i.

Rng will add new edges to the graph by repeating the following operation:

Operation: Choose u and v (u \neq v) such that Vertex v can be reached by traversing exactly three edges from Vertex u, and add an edge connecting Vertices u and v. It is not allowed to add an edge if there is already an edge connecting Vertices u and v.

Find the maximum possible number of edges that can be added.

Constraints

2 \leq N \leq 10^5

1 \leq M \leq 10^5

1 \leq A_i,B_i \leq N

The graph has no self-loops or multiple edges.

The graph is connected.

Input

Input is given from Standard Input in the following format:

N M
A_1 B_1
A_2 B_2
:
A_M B_M

Output

Find the maximum possible number of edges that can be added.

Sample Input 1

6 5
1 2
2 3
3 4
4 5
5 6

Sample Output 1

4

If we add edges as shown below, four edges can be added, and no more.

Sample Input 2

5 5
1 2
2 3
3 1
5 4
5 1

Sample Output 2

5

Five edges can be added, for example, as follows:

Add an edge connecting Vertex 5 and Vertex 3.

Add an edge connecting Vertex 5 and Vertex 2.

Add an edge connecting Vertex 4 and Vertex 1.

Add an edge connecting Vertex 4 and Vertex 2.

Add an edge connecting Vertex 4 and Vertex 3.

Solution in Python (3.4.3) (judge verdict: Runtime Error)

def main():
    import sys
    input = sys.stdin.readline
    sys.setrecursionlimit(200000)

    N,M = map(int,input().split())
    edges = [[]for _ in range(N+1)]
    for _ in range(M):
        a,b = map(int,input().split())
        edges[a].append(b)
        edges[b].append(a)

    colors = [-1]*(N+1)
    def paint(n,p,c):
        ret = 1
        if colors[n]!=-1 and colors[n]!=c:
            ret = 0
        else:
            colors[n] = c
            for nx in edges[n]:
                if nx != p:
                    ret = min(ret,paint(nx,n,(c+1)%2))
        return ret

    def choose(n,k):
        import math
        return math.factorial(n)//(math.factorial(n-k)*math.factorial(k))
    res = paint(1,0,1)
    if res == 0:
        ans = choose(N,2)-M
    else:
        ans = colors.count(0)*(N-colors.count(0))-M

    print(ans)

main()