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 (3.8.2) (judge verdict: Accepted)

import sys
from collections import deque
input = sys.stdin.readline

def main():
    n, m = map(int, input().split())
    
    tree = [[] for _ in range(n)]
    for i in range(m):
        a, b = map(int, input().split())
        tree[a-1].append(b-1)
        tree[b-1].append(a-1)
    
    judge = True
    dist = [0]*n
    already = [False]*n
    not_yet = deque([0])
    already[0] = True
    while not_yet:
        key = not_yet.popleft()
        for v in tree[key]:
            if already[v]:
                if dist[v] != (dist[key] + 1) % 2:
                    judge = False
                continue
            already[v] = True
            dist[v] = (dist[key] + 1) % 2
            not_yet.append(v)
    
    if judge:
        sub = sum(dist)
        ans = sub * (n - sub) - m
    else:
        ans = n * (n - 1) // 2 - m
    
    print(ans)
    
    
    
    
    
if __name__ == "__main__":
    main()