Problem:
Score : 800 points

Problem Statement

Takahashi has received an undirected graph with N vertices, numbered 1, 2, ..., N.
The edges in this graph are represented by (u_i, v_i).
There are no self-loops and multiple edges in this graph.

Based on this graph, Takahashi is now constructing a new graph with N^2 vertices, where each vertex is labeled with a pair of integers (a, b) (1 \leq a \leq N, 1 \leq b \leq N).
The edges in this new graph are generated by the following rule:

Span an edge between vertices (a, b) and (a', b') if and only if both of the following two edges exist in the original graph: an edge between vertices a and a', and an edge between vertices b and b'.

How many connected components are there in this new graph?

Constraints

2 \leq N \leq 100,000

0 \leq M \leq 200,000

1 \leq u_i < v_i \leq N

There exists no pair of distinct integers i and j such that u_i = u_j and v_i = v_j.

Input

The input is given from Standard Input in the following format:

N M
u_1 v_1
u_2 v_2
:
u_M v_M

Output

Print the number of the connected components in the graph constructed by Takahashi.

Sample Input 1

3 1
1 2

Sample Output 1

7

The graph constructed by Takahashi is as follows.

Sample Input 2

7 5
1 2
3 4
3 5
4 5
2 6

Sample Output 2

18

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <iostream>
#include <algorithm>
#include <string>
#include <vector>
#include <cmath>
#include <map>
#include <queue>
#include <iomanip>
#include <set>
#include <tuple>
#define mkp make_pair
#define mkt make_tuple
#define rep(i,n) for(int i = 0; i < (n); ++i)
using namespace std;
typedef long long ll;
const ll MOD=1e9+7;

vector<int> g[100010];
int used[100010];

bool dfs(int now,int col){
    used[now]=col;
    for(auto nex:g[now]){
        if(used[nex]==0){
            if(dfs(nex,-col)==false) return false;
        }else if(used[nex]==used[now]) return false;
    }
    return true;
}

int main(){
  int N,M;
  cin>>N>>M;
  rep(i,M){
      int a,b;
      cin>>a>>b;
      a--;b--;
      g[a].push_back(b);
      g[b].push_back(a);
  }

  ll ais=0;
  for(int i=0;i<N;i++) if(g[i].size()==0) ais++;

  ll com=0;
  ll ans=0;
  ll two=0;
  for(int i=0;i<N;i++){
      if(g[i].size()==0) continue;
      if(used[i]==0){
          com++;
          if(dfs(i,1)){
              ans+=2;
              two++;
          }else ans++;
      }
  }

  ans+=2*(two*(two-1));
  ans+=2*two*(com-two);
  ans+=(com-two)*(com-two-1);
  ans+=(2*ais*N-ais-(ais)*(ais-1));
  cout<<ans<<endl;



  return 0;
}