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)

# frozen_string_literal: true

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

$g = {}

$M.times do
  a, b = gets.chomp.split(' ').map(&:to_i)
  $g[a] ||= []
  $g[a] << b

  $g[b] ||= []
  $g[b] << a
end

def is_bipartite_graph(graph)
  stack = [[graph.keys.first, 1]]

  red = []
  white = []

  until stack.empty?
    puts graph.inspect
    point, color = stack.pop
    if color == 1
      return false if white.include?(point)
      red << point unless red.include?(point)
    else
      return false if red.include?(point)
      white << point unless white.include?(point)
    end
    next if graph[point].nil?
    graph[point].each do |p|
      stack << [p, (color - 1).abs]
    end
    graph.delete(point)
  end
  [red.size, white.size]
end

data = is_bipartite_graph($g)

if data
  puts data[0] * data[1] - $M
else
  puts $N * ($N - 1) / 2 - $M
end