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)

n,m = gets.chomp.split(" ").map(&:to_i);

$color = Array.new(n, 0)
$graph = []
n.times do
	$graph << []
end

m.times do
	a,b = gets.chomp.split(" ").map(&:to_i);
	$graph[a-1] << b-1
	$graph[b-1] << a-1
end

# 二部グラフか判定
def dfs(v, c, depth)
	if (depth >= $limit)
		$lastv << v
		$lastc << c
		return false
	end
	$color[v] = c;
#	puts "v = #{v} c = #{c}"
	if c == 1 then
		$plus += 1
	else
		$minus += 1
	end

	$graph[v].each do |nei|
		if $color[nei] == c then
			return false
		end
		if ($color[nei] == 0 and !dfs(nei, -c, depth+1)) then
			return false
		end
	end
return true
end

$plus = 0
$minus = 0
$limit = 2000
$lastv = []
$lastc = []

res = dfs(0, 1, 0)

while true do
	#if $plus + $minus == n then
	if $lastv.size == 0 then
		if res then
			# 二部グラフ
			puts $plus * $minus - m
		else
			# not二部グラフ
			puts n * (n-1) / 2 - m
		end
		break
	else
		v = $lastv.pop
		c = $lastc.pop
=begin
		if c == 1 then
			$plus -= 1
		else
			$minus -= 1
		end
=end

		res = dfs(v, c, 0)
	end
end