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: Wrong Answer)

ENV[Z='RUBY_THREAD_VM_STACK_SIZE']||exec({Z=>'10000000'},'ruby',$0)

N, M = gets.split.map(&:to_i)

$g = Array.new(N){ [] }

M.times do |i|
  a, b = gets.split.map(&:to_i)
  a -= 1
  b -= 1
  $g[a] << b
  # $g[b] << a
end

$color = Array.new(N, 0)

def is_bipartite_graph v, c
  
  # p [v,c]
  $color[v] = c
  
  $g[v].each do |w|
    
    return false if $color[w] == c
    return false if $color[w] == 0 && !is_bipartite_graph(w, -c)
  end
  return true
end

# p $g
# p $color
    
if is_bipartite_graph(0, 1)
  b = $color.count(1)
  w = N - b
  puts b * w - M
else
  puts N * ( N - 1 ) / 2 - M
end