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 = 4Output:4Explanation:

Input:N = 3Output:2Explanation:

If there are three edges in 3-vertex graph then it will have a triangle.

**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(n^{2}/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.

- Then, the Degree of all vertex –

- By Cauchy-Schwarz inequality –

- Therefore, 4m
^{2}/ n ≤ mn, which implies m ≤ n^{2}/ 4

Below is the implementation of above approach:

## C++

`// C++ implementation to find the maximum` `// number of edges for triangle free graph` `#include <bits/stdc++.h>` `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` |

## Javascript

`<script>` `// Javascript 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.` `function` `solve(n)` `{` ` ` ` ` `// According to the Mantel's theorem` ` ` `// the maximum number of edges will be` ` ` `// floor of [(n^2)/4]` ` ` `var` `ans = (n * n / 4);` ` ` `return` `ans;` `}` `// Driver code` `var` `n = 10;` `document.write(solve(n));` `// This code is contributed by aashish1995` `</script>` |

**Output:**

25

**Time Complexity:** O(1)

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