Maximum number of edges in Bipartite graph

Given an integer N which represents the number of Vertices. The Task is to find the maximum number of edges possible in a Bipartite graph of N vertices.

Bipartite Graph:

A Bipartite graph is one which is having 2 sets of vertices.
The set are such that the vertices in the same set will never share an edge between them.

Examples:

Input: N = 10
Output: 25
Both the sets will contain 5 vertices and every vertex of first set
will have an edge to every other vertex of the second set
i.e. total edges = 5 * 5 = 25

Input: N = 9
Output: 20

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

Approach: The number of edges will be maximum when every vertex of a given set has an edge to every other vertex of the other set i.e. edges = m * n where m and n are the number of edges in both the sets. in order to maximize the number of edges, m must be equal to or as close to n as possible. Hence, the maximum number of edges can be calculated with the formula,

Below is the implementation of the above approach:

C++

// C++ implementation of the approach 
#include <bits/stdc++.h> 
using namespace std; 
  
// Function to return the maximum number 
// of edges possible in a Bipartite 
// graph with N vertices 
int maxEdges(int N) 
{ 
    int edges = 0; 
  
    edges = floor((N * N) / 4); 
  
    return edges; 
} 
  
// Driver code 
int main() 
{ 
    int N = 5; 
    cout << maxEdges(N); 
  
    return 0; 
} 

Java

// Java implementation of the approach 
  
class GFG { 
  
    // Function to return the maximum number 
    // of edges possible in a Bipartite 
    // graph with N vertices 
    public static double maxEdges(double N) 
    { 
        double edges = 0; 
  
        edges = Math.floor((N * N) / 4); 
  
        return edges; 
    } 
  
    // Driver code 
    public static void main(String[] args) 
    { 
        double N = 5; 
        System.out.println(maxEdges(N)); 
    } 
} 
  
// This code is contributed by Naman_Garg. 

Python3

# Python3 implementation of the approach  
  
# Function to return the maximum number  
# of edges possible in a Bipartite  
# graph with N vertices  
def maxEdges(N) :  
  
    edges = 0;  
  
    edges = (N * N) // 4;  
  
    return edges;  
  
# Driver code  
if __name__ == "__main__" : 
      
    N = 5;  
    print(maxEdges(N));  
  
# This code is contributed by AnkitRai01 

C#

// C# implementation of the approach 
using System; 
  
class GFG { 
  
    // Function to return the maximum number 
    // of edges possible in a Bipartite 
    // graph with N vertices 
    static double maxEdges(double N) 
    { 
        double edges = 0; 
  
        edges = Math.Floor((N * N) / 4); 
  
        return edges; 
    } 
  
    // Driver code 
    static public void Main() 
    { 
        double N = 5; 
        Console.WriteLine(maxEdges(N)); 
    } 
} 
  
// This code is contributed by jit_t. 

PHP

<?php 
// PHP implementation of the approach 
  
// Function to return the maximum number 
// of edges possible in a Bipartite 
// graph with N vertices 
  
function maxEdges($N) 
{ 
    $edges = 0; 
  
    $edges = floor(($N * $N) / 4); 
  
    return $edges; 
} 
  
// Driver code 
    $N = 5; 
    echo maxEdges($N); 
  
// This code is contributed by ajit. 
?> 

Output:

My Personal Notes

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

C++

Java

Python3

C#

PHP

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