In a graph with N vertices, the maximum number of edges such that the graph does not contain triangles | Mantel's Theorem

The concept of triangle-free graphs is crucial to the study of graph theory, where no set of three vertices can form a triangle. It's surprising how many edges an N-vertex graph can have without triangles. Mantel's theorem provides an elegant solution to this problem. The maximum number of sides in a graph can be determined by Mantel's theorem without generating any triangles.

Method used

• Mantel algorithm

Mantel algorithm

Mantel algorithm The theorem is a famous result in graph theory that reveals how many edges a graph without triangles might have. According to this theory, if you want an N-vertex graph to be triangle-free, you cannot exceed (N * (N − 1) / 2).

Algorithm

• #Collects the N (total number of vertices) input by the user.

• We can determine the maximum number of sides by applying Mantel’s theorem.

• Maximum edge = (N * (N − 1)) / 2.

• Show as many advantages as possible to the end user.

Example

#include <iostream>

using namespace std;

// Function to calculate the maximum number of edges in a triangle-free graph using Mantel&#39;s theorem
int maxEdgesTriangleFree(int N) {
    return (N * (N - 1)) / 2;
}

int main() {
    int N;
   N=7;

    int maxEdges = maxEdgesTriangleFree(N);

    cout << "The maximum number of edges in a triangle-free graph with " << N << " vertices is: " << maxEdges << endl;

    return 0;
}

Output

The maximum number of edges in a triangle-free graph with 7 vertices is: 21

Conclusion

In short, with the help of the concept of triangle-free graph and Mantel’s theorem, we can Understand the structure and constraints of triangle-free graphs more easily. A triangle-free graph has a maximum number of sides, revealing its characteristics and practical applications.

Many fields, including network analysis, social network modeling, and algorithm creation, can benefit from this discovery. Mantel's theorem allows us to examine network connections, optimize graph algorithms, and discover novel graph architectures. This theorem also provides a springboard for further exploring the characteristics and interrelationships of graphs, paving the way for future research and development in the field of graph theory.

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