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)

fn main() {
    let s = std::io::stdin();
    let mut sc = Scanner { stdin: s.lock() };

    let n = sc.read();
    let m = sc.read();
    let mut graph = vec![vec![]; n];
    let mut uf = UnionFind::new(n);
    for _ in 0..m {
        let u = sc.read::<usize>() - 1;
        let v = sc.read::<usize>() - 1;
        graph[u].push(v);
        graph[v].push(u);
        uf.unite(u, v);
    }

    let mut color = vec![0; n];
    let mut single = 0;
    let mut bipartite = 0;
    let mut other = 0;

    for i in 0..n {
        let root = uf.find(i);
        if color[root] == 0 {
            color[root] = 1;
            let is_bipartite = color_dfs(root, &mut color, &graph);
            if uf.sizes[root] == 1 {
                single += 1;
            } else if is_bipartite {
                bipartite += 1;
            } else {
                other += 1;
            }
        }
    }

    let mut ans = 0;
    ans += single * n * 2 - single * single;
    ans += bipartite * bipartite * 2;
    ans += other * (bipartite + other) * 2 - other * other;
    println!("{}", ans);
}

fn color_dfs(v: usize, color: &mut Vec<usize>, graph: &Vec<Vec<usize>>) -> bool {
    let next_color = if color[v] == 1 { 2 } else { 1 };
    for &u in graph[v].iter() {
        if color[u] == 0 {
            color[u] = next_color;
            if !color_dfs(u, color, graph) {
                return false;
            }
        } else if color[u] != next_color {
            return false;
        }
    }
    true
}

pub struct UnionFind {
    parent: Vec<usize>,
    sizes: Vec<usize>,
    size: usize,
}

impl UnionFind {
    pub fn new(n: usize) -> UnionFind {
        UnionFind {
            parent: (0..n).map(|i| i).collect::<Vec<usize>>(),
            sizes: vec![1; n],
            size: n,
        }
    }

    pub fn find(&mut self, x: usize) -> usize {
        if x == self.parent[x] {
            x
        } else {
            let px = self.parent[x];
            self.parent[x] = self.find(px);
            self.parent[x]
        }
    }

    pub fn unite(&mut self, x: usize, y: usize) -> bool {
        let parent_x = self.find(x);
        let parent_y = self.find(y);
        if parent_x == parent_y {
            return false;
        }

        let (large, small) = if self.sizes[parent_x] < self.sizes[parent_y] {
            (parent_y, parent_x)
        } else {
            (parent_x, parent_y)
        };

        self.parent[small] = large;
        self.sizes[large] += self.sizes[small];
        self.sizes[small] = 0;
        self.size -= 1;
        return true;
    }
}

pub struct Scanner<R> {
    stdin: R,
}

impl<R: std::io::Read> Scanner<R> {
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .stdin
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}