N顶点图可以具有的最大边数，以使该图不包含三角形。曼特尔定理

给定数字N(即图中的节点数)，任务是找到N-顶点图可以具有的最大边数，以使图不包含三角形(这意味着不应有任何三个边A，图中的B，C，使得A连接到B，B连接到C，C连接到A)。该图不能包含自环或多边。

例子：

Input: N = 4
Output: 4
Explanation:

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

方法：此问题可使用曼特尔定理其中指出，在图中的边，而不含有任何三角形的最大数目是地板来解决(N ^2/4)。换句话说，必须删除几乎一半的边缘才能获得无三角形的图形。

曼特尔定理是如何工作的?
对于任何图，如果该图是无三角形的，则对于任何顶点Z只能连接到x和y中任一顶点，即对于连接在x和y之间的任何边，d(x)+ d(y)≤ N，其中d(x)和d(y)是顶点x和y的度数。

• 然后，所有顶点的度–
  
• 由柯西·舒瓦兹(Cauchy-Schwarz)不等式–
  
• 因此，4M ² / N≤MN，这意味着米≤Ñ^四分之二

下面是上述方法的实现：

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

输出：

25

时间复杂度： O(1)