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 C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
 
using namespace std;
 
using ll = long long;
 
const int INF = 1145141919;
 
vector<int> g[100005];
vector<int> color;
 
bool dfs(int cur) {
    for (int nex : g[cur]) {
        if (color[nex] == color[cur]) {
            return false;
        }
        if (color[nex] != INF) continue;
        color[nex] = !color[cur];
        if (!dfs(nex)) return false;
    }
    return true;
}
 
int main() {
    cin.tie(0);
    ios::sync_with_stdio(false);
    int n, m;
    cin >> n >> m;
    ll ans = 0;
    for (int i = 0; i < m; i++) {
        int a, b;
        cin >> a >> b;
        a--;
        b--;
        g[a].push_back(b);
        g[b].push_back(a);
    }
    color.resize(n, INF);
    color[0] = 0;
    if (dfs(0)) {
        ll cnt = 0;
        for (int i = 0; i < n; i++) cnt += color[i];
        cout << cnt * (n - cnt) - m << endl;
    } else {
        cout << (ll)n * (n - 1) / 2 - m << endl;
    }
    return 0;
}