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: Accepted)

#include <bits/stdc++.h>
using namespace std;
#define ll long long
#define all(V) V.begin(),V.end()

vector<int> G[100010];
short arr[100010];

ll cnt;

bool dfs(int c, int p, short v) {
	bool jd = 0;
	for (int i : G[c]) {
		if (i == p)continue;
		if (arr[i] == 3 - v) {
			jd = 1;
			continue;
		}
		if (arr[i] == v) {
			continue;
		}
		arr[i] = v;
		cnt++;
		if (dfs(i, c, 3 - v))jd = 1;
	}
	return jd;
}

int main() {
	ll N, M, a, b;
	cin >> N >> M;
	for (int i = 0;i < M;i++) {
		cin >> a >> b;
		a--;b--;
		G[a].push_back(b);
		G[b].push_back(a);
	}

	ll lef = N * N, ed = 0;
	ll cone = 0, alc = 0;
	bool rt;
	for (int i = 0;i < N;i++) {
		if (arr[i])continue;
		cnt = 1;
		arr[i] = 2;
		rt = dfs(i, -1, 1);
		if (cnt == 1)continue;
		if (rt) {
			alc++;
			ed += cnt;
		}
		else {
			cone++;
			alc++;
			ed += cnt;
		}
	}
	lef -= ed * ed;
	lef += alc * alc + cone * cone;
	cout << lef << endl;
}