Certainly! Here's a modified version of your Codeforces blog:

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Basic Graph Theory Short Note

1. Degree Sum

[2 \times (\text{degree counter})]

2. Handshaking Lemma

Even nodes always have an odd number of degrees. In-degree and out-degree satisfy: [ \text{Sum of in-degree} = \text{Sum of out-degree} ]

3. Calculating the Complexity of Graph

[ V = \text{vertices}, E = \text{edges} ] [ \mathcal{O}(V + E) \text{ or } \mathcal{O}(V^2) ]

4. How to Store Graph

Coming soon...

5. Adjacency Matrix

#include<bits/stdc++.h>
using namespace std;
int g[100][100];
int main() {
    int n, e; cin >> n >> e;
    while(e--) {
        int u, v; cin >> u >> v;
        g[u][v] = 1;
        g[v][u] = 1;
    }
    for(int i = 0; i < n; i++) {
        for(int j = 0; j < n; j++) {
            cout << g[i][j] << " ";
        }
        cout << endl;
    }
    if(g[4][2]) {
        cout << "YES\n";
    } else {
        cout << "NO\n";
    }
}

6. Adjacency List

#include<bits/stdc++.h>
using namespace std;
const int N = 105;
vector<int> g[N];
int indgr[N], outdgr[N];
int main() {
    int n, e; cin >> n >> e;
    while(e--) {
        int u, v; cin >> u >> v;
        indgr[v]++;
        outdgr[u]++;
        g[u].push_back(v);
    }
    for(int i = 0; i <= n; i++) {
        cout << indgr[i] << " ";
    }
    cout << "\n";
    for(int i = 0; i <= n; i++) {
        cout << outdgr[i] << " ";
    }
}

7. DFS

#include<bits/stdc++.h>
#define itsmdshahin ios_base::sync_with_stdio(false);cin.tie(0);cout.tie(0)
#define ll long long
using namespace std;
const int N = 1e5 + 9;
bool vis[N];
vector<int> g[N];
void dfs(int s) {
    vis[s] = true;
    for(auto it : g[s]) {
        if(!vis[it]) {
            cout << "VISITED " << it << endl;
            dfs(it);
        }
    }
}
int main() {
    int n, m; cin >> n >> m;
    while(m--) {
        int x, y; cin >> x >> y;
        g[x].push_back(y);
        g[y].push_back(x);
    }
    int s; cin >> s;
    dfs(s);
    for(int i = 1; i <= n; i++) {
        if(!vis[i]) {
            cout << "DISCONNECTED GRAPH\n";
            return 0;
        }
    }
}

8. Connected Components

#include<bits/stdc++.h>
#define itsmdshahin ios_base::sync_with_stdio(false);cin.tie(0);cout.tie(0)
#define ll long long
using namespace std;
const int N = 1e5 + 9;
bool vis[N];
vector<int> g[N];
void dfs(int s) {
    vis[s] = true;
    for(auto it : g[s]) {
        if(!vis[it]) {
            dfs(it);
        }
    }
}

int main() {
    int n, m; cin >> n >> m;
    while(m--) {
        int x, y; cin >> x >> y;
        g[x].push_back(y);
        g[y].push_back(x);
    }
    int ans = 0;
    for(int i = 1; i <= n; i++) {
        if(!vis[i]) {
            dfs(i);
            ++ans;
        }
    }
    cout << "Connected Components: " << ans << endl;
}

9. BFS Implementation

#include<bits/stdc++.h>
using namespace std;
const int N = 1e5 + 9;
vector<int> G[N];
bool Vis[N];
int main() {
    int n, e; cin >> n >> e;
    while(e--) {
        int x, y; cin >> x >> y;
        G[x].push_back(y);
        G[y].push_back(x);
    }
    queue<int> q;
    q.push(1);
    Vis[1] = true;
    while(!q.empty()) {
        int u = q.front();
        q.pop();
        for(auto it : G[u]) {
            if(!Vis[it]) {
                q.push(it);
                Vis[v] = true;
            }
        }
    }
}

10. BFS Component Find in Graph

#include<bits/stdc++.h>
using namespace std;
const int N = 1e5 + 9;
vector<int> G[N];
bool Vis[N];
queue<int> q;
void bfs(int n) {
    q.push(n);
    Vis[n] = true;
    while(!q.empty()) {
        int u = q.front();
        q.pop();
        for(auto it : G[u]) {
            if(!Vis[it]) {
                q.push(it);
                Vis[it] = true;
            }
        }
    }
}

int main() {
    int n, e; cin >> n >> e;
    int ans = 0;
    for(int i = 0; i < n; i++) {
        if(!Vis[i]) {
            bfs(i);
            ++ans;
        }
    }
    cout << ans << endl;
}

11. BFS Distance Find in Graph Theory

#include<bits/stdc++.h>
using namespace std;
const int N = 1e5 + 9;
vector<int> G[N];
bool Vis[N];
vector<int> dis(N);
int main() {
    int n, e; cin >> n >> e;
    queue<int> q;
    q.push(1);
    Vis[1] = true;
    dis[1] = 0;
    while(!q.empty()) {
        int u = q.front();
        q.pop();
        for(auto it : G[u]) {
            if(!Vis[it]) {
                q.push(it);
                dis[it] = dis[u] + 1;
                Vis[it] = true;
            }
        }
    }
    for(int i = 1; i <= n; i++) {
        cout << dis[i] << " ";
    }
    cout << endl;
}

12. BFS Path Find in Graph Theory

Coming soon...

13. Bicoloring and Bipartite Graph

#include<bits/stdc++.h>
using namespace std;
const int N = 105;
vector<int> g[N];
bool vis[N], col[N

], ok;

void dfs(int u) {
    vis[u] = true;
    for(auto v : g[u]) {
        if(!vis[v]) {
            col[v] = col[u] ^ 1;
            dfs(v);
        } else {
            if(col[u] == col[v]) ok = false;
        }
    }
}

int main() {
    int n, e;
    cin >> n >> e;
    while(e--) {
        int u, v;
        cin >> u >> v;
        g[u].push_back(v);
        g[v].push_back(u);
    }
    ok = true;
    for(int i = 1; i <= n; i++) {
        if(!vis[i]) dfs(i);
    }
    if(ok) {
        cout << "YES\n";
    } else {
        cout << "NO\n";
    }
}

Coming soon...