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;
typedef long long ll;

ll N, M;
vector<vector<ll>> G;
vector<ll> colors;
//頂点vから辺をちょうど3本たどる
bool is_bipartite(ll v, ll color = 0) {
    colors[v] = color;
    for (auto nv : G[v]) {
        if (colors[nv] == color) return false;
        else if (colors[nv] == -1 && !(is_bipartite(nv, 1-color))) {
            return false;
        }
    }
    return true;
}

int main() {
    cin >> N >> M;
    G.resize(N);
    colors.assign(N, -1);
    for (int i = 0; i < M; i++) {
        ll a, b; cin >> a >> b; a--; b--;
        G[a].push_back(b);
        G[b].push_back(a);
    }
    if (is_bipartite(0)) {
        ll B, W = 0;
        for (int i = 0; i < N; i++) {
            W += colors[i];
        }
        B = N-W;
        cout << B*W-M << endl;
    }
    else cout << N*(N-1)/2-M << endl;
}