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 C (GCC 9.2.1) (judge verdict: Accepted)

#include <stdio.h>
#include <stdlib.h>
int *tree[100000];
int cnt[100000], cnts[100000];
int colors[100000];
long long int black = 0, white = 0;
int dfs(int v, int color){
    colors[v] = color;
    if(color == 1){
        white++;
    }else if(color == -1){
        black++;
    }
    for(int i = 0; i < cnt[v]; i++){
        int v_i = tree[v][i];
        if(colors[v_i] == color){
            return 0;
        }
        if(colors[v_i] == 0 && dfs(v_i, -color) == 0){
            return 0;
        }
    }
    return 1;
}
int main(void){
    long long int n, m;
    scanf("%lld%lld", &n, &m);
    int a[100000], b[100000];
    int i, j;
    for(i = 0; i < n; i++){
        cnt[i] = 0;
        cnts[i] = 0;
        colors[i] = 0;//0:tbd -1:b 1:w
    }
    for(i = 0; i < m; i++){
        scanf("%d%d", &a[i], &b[i]);
        a[i]--;
        b[i]--;
        cnt[a[i]]++;
        cnt[b[i]]++;
    }
    for(i = 0; i < n; i++){
        tree[i] = malloc(sizeof(int) * (cnt[i] + 1));
    }
    for(i = 0; i < m; i++){
        tree[a[i]][cnts[a[i]]++] = b[i];
        tree[b[i]][cnts[b[i]]++] = a[i];
    }

    if(dfs(0, 1) == 0){
        printf("%lld\n", n*(n-1)/2 - m);
        
    }else{
        printf("%lld\n", black*white-m);
    }
    return 0;
}