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 PyPy3 (2.4.0) (judge verdict: Accepted)

from collections import defaultdict, deque

N, M = map(int, input().split())

dic = defaultdict(set)
for _ in range(M):
    A, B = map(int, input().split())
    dic[A].add(B)
    dic[B].add(A)


colors = [0]*(N+1)

def dfs():
    que = deque([(1, 1)])

    while len(que):
        v, color = que.popleft()
        colors[v] = color
        for to in list(dic[v]):
            if (color == colors[to]):
                return False
            if (colors[to] == 0):
                que.append((to, -color))
                
    return True


flag = dfs()
b = colors.count(1)

print(b*(N-b) - M if (flag) else N*(N-1)//2 - M)