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: Time Limit Exceeded)

from queue import LifoQueue
N,M = map(int, input().split())
es = [[] for i in range(N)]
for i in range(M):
    a,b = map(int, input().split())
    a,b = a-1,b-1
    es[a].append(b)
    es[b].append(a)

colors = [0 for i in range(N)]

def is_bipartite():
    stack = LifoQueue()
    stack.put([0, 1])
    while not stack.empty():
        v, color = stack.get()
        colors[v] = color
        for to in es[v]:
            if colors[to] == color:
                 return False
            if colors[to] == 0:
                 stack.put([to, -color])
    return True

if is_bipartite():
    b = (sum(colors) + N) // 2
    w = N-b
    print(b*w - M)
else:
    all = N*(N-1) // 2
    print(all - M)