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 Rust (1.15.1) (judge verdict: Accepted)

fn main() {
  let (n, m): (usize, usize) = read_tuple();
  let edges: Vec<(usize, usize)> = read_tuples(m);

  let mut graph: Vec<Vec<usize>> = vec![vec![]; n+1];
  for (v,w) in edges {
    graph[v].push(w);
    graph[w].push(v);
  }

  match is_bipartite(&graph) {
    (p, _) if !p     => {println!("{}", n * (n-1) / 2 - m)},
    (_, Some((b,w))) => {println!("{}", b*w - m)},
    _ => unreachable!()
  }
}

fn is_bipartite(g: &Vec<Vec<usize>>) -> (bool, Option<(usize, usize)>) {
  fn dfs(v: usize, p: bool, g: &Vec<Vec<usize>>, cv: &mut Vec<Option<bool>>) -> bool {
    match cv[v] {
      None => {
        let next = &g[v];
        cv[v] = Some(p);
        next.iter().all(|&w|dfs(w,!p,g,cv))
      },
      Some(q) if q == p => {true},
      Some(q) if q != p => {false},
      _ => unreachable!()
    }
  }

  let mut canvas: Vec<Option<bool>> = vec![None; g.len()];
  let ans = dfs(1, false, g, &mut canvas);
  let b = canvas.iter().filter(|&&v|v==Some(false)).count();
  let w = canvas.iter().filter(|&&v|v==Some(true)).count();
  (ans, if ans {Some((b,w))} else {None})
}

fn read_tuple<T1: std::str::FromStr, T2: std::str::FromStr>() -> (T1, T2) {
  let mut buf = String::new();
  std::io::stdin().read_line(&mut buf).ok();
  let mut it = buf.trim().split_whitespace();
  let x = it.next().unwrap().parse::<T1>().ok().unwrap();
  let y = it.next().unwrap().parse::<T2>().ok().unwrap();
  (x, y)
}

fn read_tuples<T1: std::str::FromStr, T2: std::str::FromStr>(n: usize) -> Vec<(T1, T2)> {
  let mut vec: Vec<(T1, T2)> = vec![];
  for _ in 0 .. n {
    vec.push(read_tuple());
  }
  vec
}