# Maximum number of edges that N-vertex graph can have such that graph is Triangle free | Mantel’s Theorem

Given a number N which is the number of nodes in a graph, the task is to find the maximum number of edges that N-vertex graph can have such that graph is triangle-free (which means there should not be any three edges A, B, C in the graph such that A is connected to B, B is connected to C and C is connected to A). The graph cannot contain a self-loop or multi edges.

Examples:

Input: N = 3
Output: 2
Explanation:
If there are three edges in 3-vertex graph then it will have a triangle.

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Approach: This Problem can be solved using Mantel’s Theorem which states that the maximum number of edges in a graph without containing any triangle is floor(n2/4). In other words, one must delete nearly half of the edges to obtain a triangle-free graph.

How Mantel’s Theorem Works ?
For any Graph, such that the graph is Triangle free then for any vertex Z can only be connected to any of one vertex from x and y, i.e. For any edge connected between x and y, d(x) + d(y) ≤ N, where d(x) and d(y) is the degree of the vertex x and y.

Below is the implementation of above approach:

## C++

 `// C++ implementation to find the maximum ` `// number of edges for triangle free graph ` ` `  `#include ` `using` `namespace` `std; ` ` `  `// Function to find the maximum number of ` `// edges in a N-vertex graph. ` `int` `solve(``int` `n) ` `{ ` `    ``// According to the Mantel's theorem ` `    ``// the maximum number of edges will be ` `    ``// floor of [(n^2)/4] ` `    ``int` `ans = (n * n / 4); ` ` `  `    ``return` `ans; ` `} ` ` `  `// Driver Function ` `int` `main() ` `{ ` `    ``int` `n = 10; ` `    ``cout << solve(n) << endl; ` `    ``return` `0; ` `} `

## Java

 `// Java implementation to find the maximum  ` `// number of edges for triangle free graph  ` `class` `GFG  ` `{  ` ` `  `    ``// Function to find the maximum number of  ` `    ``// edges in a N-vertex graph.  ` `    ``public` `static` `int` `solve(``int` `n)  ` `    ``{ ` `         `  `        ``// According to the Mantel's theorem  ` `        ``// the maximum number of edges will be  ` `        ``// floor of [(n^2)/4]  ` `        ``int` `ans = (n * n / ``4``);  ` ` `  `        ``return` `ans;  ` `    ``}  ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `main(String args[]) ` `    ``{  ` `        ``int` `n = ``10``;  ` `        ``System.out.println(solve(n));  ` `    ``}  ` `} ` ` `  `// This code is contributed by divyamohan123 `

## C#

 `// C# implementation to find the maximum  ` `// number of edges for triangle free graph  ` `using` `System; ` ` `  `class` `GFG  ` `{  ` ` `  `    ``// Function to find the maximum number of  ` `    ``// edges in a N-vertex graph.  ` `    ``public` `static` `int` `solve(``int` `n)  ` `    ``{ ` `         `  `        ``// According to the Mantel's theorem  ` `        ``// the maximum number of edges will be  ` `        ``// floor of [(n^2)/4]  ` `        ``int` `ans = (n * n / 4);  ` ` `  `        ``return` `ans;  ` `    ``}  ` ` `  `    ``// Driver code ` `    ``public` `static` `void` `Main() ` `    ``{  ` `        ``int` `n = 10;  ` `        ``Console.WriteLine(solve(n));  ` `    ``}  ` `} ` ` `  `// This code is contributed by AnkitRai01 `

## Python3

 `# Python3 implementation to find the maximum ` `# number of edges for triangle free graph ` ` `  `# Function to find the maximum number of ` `# edges in a N-vertex graph. ` `def` `solve(n): ` `     `  `    ``# According to the Mantel's theorem ` `    ``# the maximum number of edges will be ` `    ``# floor of [(n^2)/4] ` `    ``ans ``=` `(n ``*` `n ``/``/` `4``) ` ` `  `    ``return` `ans ` ` `  `# Driver Function ` `if` `__name__ ``=``=` `'__main__'``: ` `    ``n ``=` `10` `    ``print``(solve(n)) ` ` `  `# This code is contributed by mohit kumar 29 `

Output:

```25
```

Time Complexity: O(1)

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.

My Personal Notes arrow_drop_up Third year Department of Information Technology Jadavpur University

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.