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 Ruby (2.3.3) (judge verdict: Accepted)

if !ENV['RUBY_THREAD_VM_STACK_SIZE']
  exec({'RUBY_THREAD_VM_STACK_SIZE'=>'1000000000'}, '/usr/bin/ruby', $0)
else
  class Hash
    def bipartite_graph?
      color = Hash.new(0)

      dfs(1, 1, color)
    end

    private

    def dfs(v, c, color)
      color[v] = c

      E[v].each do |u, _|
        return false if color[u] == c
        return false if color[u] == 0 && !dfs(u, -c, color)
      end

      return true
    end
  end

  N, M = gets.split.map(&:to_i)
  E = Hash.new { |h, k| h[k] = {} }
  edges = M.times.map { gets.split.map(&:to_i) }

  edges.each do |a, b|
    E[a][b] = true
    E[b][a] = true
  end

  if E.bipartite_graph?
    color = Hash.new(0)

    dfs = ->(v, c) {
      color[v] = c

      E[v].each do |u, _|
        next if color[u] != 0

        dfs[u, -c]
      end
    }

    dfs[1, 1]

    counter = Hash.new(0)

    color.each do |v, c|
      counter[c] += 1
    end

    puts counter.values.inject(:*) - M
  else
    puts (N * (N - 1)) / 2 - M
  end
end