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 <algorithm>
#include <iostream>
#include <vector>
using namespace std;
using ll = long long;

vector<vector<int>> g;
vector<int> color;

bool is_bipartite(int u) {
    for (int v: g[u]) {
        if (color[v] != -1) {
            if (color[v] == color[u]) return false;
        } else {
            color[v] = color[u] ^ 1;
            if (!is_bipartite(v)) return false;
        }
    }
    return true;
}

int main() {
    int V, E; cin >> V >> E;
    g.resize(V);
    for (int i = 0; i < E; i++) {
        int a, b; cin >> a >> b; a--; b--;
        g[a].emplace_back(b);
        g[b].emplace_back(a);
    }
    color.assign(V, -1);
    color[0] = 0;
    ll ans = 0;
    if (!is_bipartite(0)) {
        ans = 1LL * V * (V - 1) / 2;
    } else {
        for (int i = 0; i < V; i++) ans += color[i];
        ans *= V - ans;
    }
    ans -= E;
    cout << ans << endl;
    return 0;
}