N-顶点图可以具有的最大边数，使得图没有三角形 |曼特尔定理

给定一个数字N ，它是图中节点的数量，任务是找到 N 顶点图可以具有的最大边数，使得图没有三角形(这意味着不应该有任何三个边 A，图中的 B、C 使得 A 连接到 B，B 连接到 C，C 连接到 A)。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：这个问题可以使用曼特尔定理来解决，该定理指出图中不包含任何三角形的最大边数是 floor(n ² /4)。换句话说，必须删除近一半的边才能获得无三角形图。

曼特尔定理如何运作?
对于任何图，使得该图是无三角形的，那么对于任何顶点 Z 只能连接到来自 x 和 y 的任何一个顶点，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。

• 那么，所有顶点的度数——

• 通过柯西-施瓦茨不等式——

• 因此，4m ² / n ≤ mn，这意味着m ≤ n ² / 4

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

Javascript

 
// 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
 

输出：

25

时间复杂度： O(1)

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