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# (Mono 4.6.2.0) (judge verdict: Wrong Answer)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using static System.Console;
using static System.Convert;
using static System.Math;
using static Extentions;

class IO
{
    int idx;
    string[] input = In.ReadToEnd().Split(new[] { " ", "\n", "\r" },
        StringSplitOptions.RemoveEmptyEntries);

    T Get<T>(Func<string, T> parser) => parser(input[idx++]);

    public string S => Get(s => s);
    public char C => Get(char.Parse);
    public int I => Get(int.Parse);
    public long L => Get(long.Parse);
    public double F => Get(double.Parse);
    public decimal D => Get(decimal.Parse);
    public BigInteger B => Get(BigInteger.Parse);

    T[] Gets<T>(int n, Func<string, T> parser)
        => input.Skip((idx += n) - n).Take(n).Select(parser).ToArray();

    public string[] Ss(int n) => Gets(n, s => s);
    public char[] Cs(int n) => Gets(n, char.Parse);
    public int[] Is(int n) => Gets(n, int.Parse);
    public long[] Ls(int n) => Gets(n, long.Parse);
    public double[] Fs(int n) => Gets(n, double.Parse);
    public decimal[] Ds(int n) => Gets(n, decimal.Parse);
    public BigInteger[] Bs(int n) => Gets(n, BigInteger.Parse);

    public void Write<T>(params T[] xs) => WriteLine(string.Join(" ", xs));
    public void Write(params object[] xs) => WriteLine(string.Join(" ", xs));
}

static class Extentions
{
}

static class Program
{
    public static void Main()
    {
        var sw = new StreamWriter(OpenStandardOutput()) { NewLine = "\n" };
#if DEBUG
        sw.AutoFlush = true;
#else
        sw.AutoFlush = false;
#endif
        SetOut(sw);
        Solve(new IO());
        Out.Flush();
    }

    static void Solve(IO io)
    {
        var n = io.L;
        var m = io.L;

        var grl = new List<int>[n];
        var grs = new HashSet<int>[n];
        for (var i = 0; i < n; i++) grl[i] = new List<int>();
        for (var i = 0; i < n; i++) grs[i] = new HashSet<int>();

        for (var i = 0; i < m; i++)
        {
            var a = io.I - 1;
            var b = io.I - 1;
            grl[a].Add(b);
            grl[b].Add(a);
            grs[a].Add(b);
            grs[b].Add(a);
        }

        var hasc = false;

        for (var i = 0; i < n; i++)
            foreach (var j in grl[i])
                if (grl[j].Any(grs[i].Contains)) { hasc = true; goto ExitLoop; }
        ExitLoop:;

        var max = grl.Max(x => x.Count);

        if (n == 2)
            io.Write(0);
        else if (n == 3)
            io.Write(3 - m);
        else if (n == 4)
            io.Write(hasc ? 6 - m : 4 - m);
        else if (max == n - 1 && m == n - 1)
            io.Write(0);
        else if (n % 2 == 0 && max <= (n - 5) / 2 + 3)
            io.Write(n + ((n - 5) / 2 + 1) * n / 2 - m);
        else
            io.Write(n + ((n - 3) * n / 2) - m);
    }
}