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 (1.42.0) (judge verdict: Accepted)

use petgraph::algo::dijkstra;
use petgraph::graph::*;
use proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;
#[allow(unused_imports)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::*;
#[allow(unused_imports)]
use std::f64::consts::*;

#[allow(unused)]
const INF: usize = std::usize::MAX / 4;
#[allow(unused)]
const M: usize = 1000000007;

fn main() {
    input! {
        n: usize,
        m: usize,
        ab: [(Usize1, Usize1); m],
    }
    let mut graph = Graph::new_undirected();
    for _ in 0..n {
        graph.add_node(());
    }
    for &(ai, bi) in &ab {
        graph.add_edge(node_index(ai), node_index(bi), ());
    }
    let distance = dijkstra(&graph, node_index(0), None, |_| 1);
    let mut color = vec![0; n];
    for i in 0..n {
        if distance[&node_index(i)] % 2 == 0 {
            color[i] = 1;
        }
    }
    if ab.iter().any(|&(ai, bi)| color[ai] == color[bi]) {
        println!("{}", n * (n - 1) / 2 - m);
    } else {
        let mut count = [0, 0];
        for i in 0..n {
            count[color[i]] += 1;
        }
        println!("{}", count[0] * count[1] - m);
    }
}