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++ (GCC 9.2.1) (judge verdict: Wrong Answer)

/**
 ____ ____ ____ ____ ____
||a |||t |||o |||d |||o ||
||__|||__|||__|||__|||__||
|/__\|/__\|/__\|/__\|/__\|

**/

#include <bits/stdc++.h>

using namespace std;

typedef long long ll;

const int N_MAX = 100002;

int n, m;

vector <int> edges[N_MAX];

bool visited[N_MAX];

bool color[N_MAX];

bool oddCycle;

int cnt;

void dfs (int u, int last = -1)
{
    visited[u] = true;
    for(int v : edges[u])
        if(visited[v] == false)
        {
            color[v] = !color[u];
            dfs(v, u);
        }
        else if(color[v] == color[u])
            oddCycle = true;
}

int main()
{
    ios_base::sync_with_stdio(false);
    cin.tie(0);
    cout.tie(0);
    cin >> n >> m;
    for(int i = 1; i <= m; i++)
    {
        int u, v;
        cin >> u >> v;
        edges[u].push_back(v);
        edges[v].push_back(u);
    }
    ll cntTrees = 0, cntGraphs = 0, cntSingle = 0;
    for(int i = 1; i <= n; i++)
        if(visited[i] == false)
        {
            oddCycle = false;
            cnt = 0;
            dfs(i);
            if(cnt == 1)
                cntSingle++;
            else if(oddCycle == false)
                cntTrees++;
            else
                cntGraphs++;
        }
    cout << (cntTrees * 2 + cntGraphs) * cntTrees + (cntTrees + cntGraphs) * cntGraphs + cntSingle * n * 2 - cntSingle * cntSingle << "\n";
    return 0;
}