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: Time Limit Exceeded)

#!/usr/bin/env python3
import sys
import collections

sys.setrecursionlimit(10 ** 7)

# 1 ✕ N
N, M = list(map(int, input().rstrip().split()))

# N ✕ M
A = list(tuple(map(int, input().rstrip().split())) for _ in range(M))

items = [set() for _ in range(N)]
for i in range(M):
    a, b = A[i]
    items[a - 1].add(b - 1)
    items[b - 1].add(a - 1)

# print(items)

init = 0
for i in items:
    init += len(i)

# print(ans)
while True:
    found = False
    for n in range(N):
        # print("-----")

        t = items[n]
        # print(t)
        k = set()
        for i in t:
            k.update(items[i])

        # print(k)
        k2 = set()
        for i in k:
            k2.update(items[i])

        # print(k2)
        if n in k2:
            k2.remove(n)

        if t != k2:
            items[n] = k2
            found = True

    if not found:
        break

end = 0
for i in items:
    end += len(i)
print(int((end - init) / 2))
# print(len(items))
# print((len(items) - ans) / 2)