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

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:

1. The Number of Cycles in the Wheel Graph.
2. A number of edges in Wheel Graph.
3. 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)