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)

import sys
sys.setrecursionlimit(100000)
N, M = map(int, input().split())
edges = [list(map(int, input().split())) for _ in range(M)]

graph = [[] for _ in range(N)]
for x, y in edges:
    graph[x - 1].append(y - 1)
    graph[y - 1].append(x - 1)

def dfs(v, c):
    # c: color = 1 or -1
    node[v] = c
    for i in graph[v]:
        if node[i] == c:
            return False
        
        if node[i] == 0 and not dfs(i, -c):
            return False
        
    return True

node = [0] * N
if dfs(0, 1):
    x = sum(v + 1 for v in node) // 2
    print(x * (N - x) - M)
else:
    print(N * (N - 1) // 2 - M)