Program to find the diameter, cycles and edges of a Wheel Graph

Namachila 2 days ago

0 0 5 minutes read

Wheel Graph: A Wheel graph is a graph formed by connecting a single universal vertex to all vertices of a cycle.

Properties:

Wheel graphs are Planar graphs.
There is always a Hamiltonian cycle in the Wheel graph.
Chromatic Number is 3 and 4, if n is odd and even respectively.

Problem Statement:

Given the Number of Vertices in a Wheel Graph. The task is to find:

The Number of Cycles in the Wheel Graph.
A number of edges in Wheel Graph.
The diameter of a Wheel Graph.

Examples:

Input: vertices = 4
Output: Number of cycle = 7
         Number of edge = 6
         Diameter = 1

Input: vertices = 6
Output: Number of cycle = 21
         Number of edge = 10
         Diameter = 2

Example #1: For vertices = 4 Wheel Graph, total cycle is 7:

Example #2: For vertices = 5 and 7 Wheel Graph Number of edges = 8 and 12 respectively:

Example #3: For vertices = 4, the Diameter is 1 as We can go from any vertices to any vertices by covering only 1 edge.

Formula to calculate the cycles, edges and diameter:

Number of Cycle = (vertices * vertices) - (3 * vertices) + 3
Number of edge = 2 * (vertices - 1)
Diameter = if vertices = 4, Diameter = 1
           if vertices > 4, Diameter = 2

Below is the required implementation:

C++

// C++ Program to find the diameter, 
// cycles and edges of a Wheel Graph
#include <bits/stdc++.h>
using namespace std;
 
// Function that calculates the
// Number of Cycle in Wheel Graph.
int totalCycle(int vertices)
{
    int result = 0;
 
    // calculates no. of Cycle.
    result = pow(vertices, 2) - (3 * vertices) + 3;
 
    return result;
}
 
// Function that calculates the
// Number of Edges in Wheel graph.
int Edges(int vertices)
{
    int result = 0;
 
    result = 2 * (vertices - 1);
 
    return result;
}
 
// Function that calculates the
// Diameter in Wheel Graph.
int Diameter(int vertices)
{
    int result = 0;
 
    // calculates Diameter.
    if (vertices == 4)
        result = 1;
    else
        result = 2;
 
    return result;
}
 
// Driver Code
int main()
{
    int vertices = 4;
 
    cout << "Number of Cycle = " << totalCycle(vertices) << endl;
    cout << "Number of Edges = " << Edges(vertices) << endl;
    cout << "Diameter = " << Diameter(vertices);
 
    return 0;
}

Java

//Java Program to find the diameter, 
// cycles and edges of a Wheel Graph
import java.io.*;
 
class GFG
{
    // Function that calculates the 
    // Number of Cycle in Wheel Graph. 
    static int totalCycle(double vertices) 
    { 
        double result = 0;
        int result1 = 0;
       
        // calculates no. of Cycle. 
        result = Math.pow(vertices, 2) - (3 * vertices) + 3; 
       
        result1 = (int)(result);
        return result1; 
    } 
       
    // Function that calculates the 
    // Number of Edges in Wheel graph. 
    static int Edges(int vertices) 
    { 
        int result = 0; 
       
        result = 2 * (vertices - 1); 
       
        return result; 
    } 
       
    // Function that calculates the 
    // Diameter in Wheel Graph. 
    static int Diameter(int vertices) 
    { 
        int result = 0; 
       
        // calculates Diameter. 
        if (vertices == 4) 
            result = 1; 
        else
            result = 2; 
       
        return result; 
    }
     
    //Driver Code
    public static void main(String[] args)
    {
        int vertices = 4;
         
        System.out.println("Number of Cycle = " + totalCycle(vertices));
        System.out.println("Number of Edges = " + Edges(vertices));
        System.out.println("Diameter = " + Diameter(vertices));
    }
}

Python3

# Python3 Program to find the diameter, 
# cycles and edges of a Wheel Graph 
 
# Function that calculates the 
# Number of Cycle in Wheel Graph. 
def totalCycle(vertices): 
 
    result = 0
 
    # calculates no. of Cycle. 
    result = (pow(vertices, 2) -
             (3 * vertices) + 3)
    return result 
 
# Function that calculates the 
# Number of Edges in Wheel graph. 
def Edges(vertices): 
 
    result = 0
    result = 2 * (vertices - 1) 
    return result 
 
# Function that calculates the 
# Diameter in Wheel Graph. 
def Diameter(vertices): 
 
    result = 0
 
    # calculates Diameter. 
    if vertices == 4: 
        result = 1
    else:
        result = 2
 
    return result 
 
# Driver Code 
if __name__ == "__main__":
 
    vertices = 4
 
    print("Number of Cycle =", 
           totalCycle(vertices)) 
    print("Number of Edges =", Edges(vertices)) 
    print("Diameter =", Diameter(vertices)) 
 
# This code is contributed by Rituraj Jain

C#

// C# Program to find the diameter, 
// cycles and edges of a Wheel Graph
using System;
class GFG
{
// Function that calculates the 
// Number of Cycle in Wheel Graph. 
static int totalCycle(double vertices) 
{ 
    double result = 0;
    int result1 = 0;
 
    // calculates no. of Cycle. 
    result = Math.Pow(vertices, 2) -
                     (3 * vertices) + 3; 
 
    result1 = (int)(result);
    return result1; 
} 
 
// Function that calculates the 
// Number of Edges in Wheel graph. 
static int Edges(int vertices) 
{ 
    int result = 0; 
 
    result = 2 * (vertices - 1); 
 
    return result; 
} 
 
// Function that calculates the 
// Diameter in Wheel Graph. 
static int Diameter(int vertices) 
{ 
    int result = 0; 
 
    // calculates Diameter. 
    if (vertices == 4) 
        result = 1; 
    else
        result = 2; 
 
    return result; 
}
 
// Driver Code
public static void Main()
{
    int vertices = 4;
     
    Console.WriteLine("Number of Cycle = " + 
                      totalCycle(vertices));
    Console.WriteLine("Number of Edges = " + 
                           Edges(vertices));
    Console.WriteLine("Diameter = " + 
                       Diameter(vertices));
}
}
 
// This code is contributed by inder_verma

PHP

<?php
// PHP Program to find the diameter, 
// cycles and edges of a Wheel Graph
 
// Function that calculates the
// Number of Cycle in Wheel Graph.
function totalCycle($vertices)
{
    $result = 0;
 
    // calculates no. of Cycle.
    $result = pow($vertices, 2) - 
                 (3 * $vertices) + 3;
 
    return $result;
}
 
// Function that calculates the
// Number of Edges in Wheel graph.
function Edges($vertices)
{
    $result = 0;
 
    $result = 2 * ($vertices - 1);
 
    return $result;
}
 
// Function that calculates the
// Diameter in Wheel Graph.
function Diameter($vertices)
{
    $result = 0;
 
    // calculates Diameter.
    if ($vertices == 4)
        $result = 1;
    else
        $result = 2;
 
    return $result;
}
 
// Driver Code
$vertices = 4;
 
echo "Number of Cycle = " , 
      totalCycle($vertices), "\n";
echo "Number of Edges = " , 
      Edges($vertices), "\n" ;
echo "Diameter = " , Diameter($vertices);
 
// This code is contributed by inder_verma
?>

Javascript

// Javascript Program to find the diameter, 
// cycles and edges of a Wheel Graph
 
// Function that calculates the
// Number of Cycle in Wheel Graph.
function totalCycle(vertices)
{
    var result = 0;
 
    // calculates no. of Cycle.
    result = Math.pow(vertices, 2) - (3 * vertices) + 3;
 
    return result;
}
 
// Function that calculates the
// Number of Edges in Wheel graph.
function Edges(vertices)
{
    var result = 0;
 
    result = 2 * (vertices - 1);
 
    return result;
}
 
// Function that calculates the
// Diameter in Wheel Graph.
function Diameter(vertices)
{
    var result = 0;
 
    // calculates Diameter.
    if (vertices === 4)
        result = 1;
    else
        result = 2;
 
    return result;
}
 
// Driver Code
 
    var vertices = 4;
 
    console.log( "Number of Cycle = " + totalCycle(vertices) + "\n");
    console.log( "Number of Edges = " + Edges(vertices) + "\n");
    console.log( "Diameter = " + Diameter(vertices));
 
// This code is contributed by Abhijeet Kumar(abhijeet19403)

Output

Number of Cycle = 7
Number of Edges = 6
Diameter = 1

Time complexity: O(1)
Auxiliary Space: O(1)

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