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 Rust (1.15.1) (judge verdict: Accepted)

//https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8 より
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut iter = $s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
    ($($r:tt)*) => {
        let s = {
            use std::io::Read;
            let mut s = String::new();
            std::io::stdin().read_to_string(&mut s).unwrap();
            s
        };
        let mut iter = s.split_whitespace();
        input_inner!{iter, $($r)*}
    };
}

macro_rules! input_inner {
    ($iter:expr) => {};
    ($iter:expr, ) => {};
    ($iter:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($iter, $t);
        input_inner!{$iter $($r)*}
    };
}

macro_rules! read_value {
    ($iter:expr, ( $($t:tt),* )) => {
        ( $(read_value!($iter, $t)),* )
    };
    ($iter:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($iter, $t)).collect::<Vec<_>>()
    };
    ($iter:expr, chars) => {
        read_value!($iter, String).chars().collect::<Vec<char>>()
    };
    ($iter:expr, usize1) => {
        read_value!($iter, usize) - 1
    };
    ($iter:expr, $t:ty) => {
        $iter.next().unwrap().parse::<$t>().expect("Parse error")
    };
}

//

fn run() {
    input! {
        n: usize,
        m: usize,
        e: [(usize1, usize1); m],
    }
    let mut g = vec![vec![]; n];
    for (a, b) in e {
        g[a].push(b);
        g[b].push(a);
    }
    let mut x = 0;
    let mut y = 0;
    let mut z = 0;
    let mut used = vec![false; n];
    let mut depth = vec![0; n];
    for r in 0..n {
        if used[r] {
            continue;
        }
        used[r] = true;
        if g[r].is_empty() {
            z += 1;
            continue;
        }
        let mut q = vec![];
        let mut stack = vec![r];
        while let Some(v) = stack.pop() {
            q.push(v);
            for &u in &g[v] {
                if used[u] {
                    continue;
                }
                used[u] = true;
                depth[u] = depth[v] ^ 1;
                stack.push(u);
            }
        }
        let mut cond = true;
        for &v in &q {
            for &u in &g[v] {
                if depth[u] == depth[v] {
                    cond = false;
                }
            }
        }
        if cond {
            x += 1;
        } else {
            y += 1;
        }
    }
    let mut ans = 0;
    ans += z * z;
    ans += 2 * (n - z) * z;
    ans += x * x * 2;
    ans += y * y;
    ans += 2 * x * y;
    println!("{}", ans);
}

fn main() {
    run();
}