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Maximum number of edges that N-vertex graph can have such that graph is Triangle free| Mantel’s Theorem
The concept of a triangle−free graph, in which no collection of three vertices forms a triangle, is crucial to the study of graph theory. It's amazing to consider how many edges an N−vertex graph may have and yet be triangle−free. Mantel's theorem offers the elegant solution to this issue.The maximum number of edges in a graph may be determined via Mantel's theorem without generating any triangles.
Methods Used
Mantel’s algorithm
Mantel’s Algorithm
Mantel's theorem is a famous conclusion in graph theory that sheds light on how many edges a graph without triangles may have. According to this theory, if you want your N−vertex graph to be triangle−free, you can't have more than (N * (N − 1) / 2).
Algorithm
Collect the user's input for N, the total number of vertices.
We can determine max no of edges by applying Mantel's theorem.
Max edge= (N * (N − 1)) / 2.
Exhibit as many edges 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'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 conclusion, understanding the structure and constraints of triangle−free graphs is made easier with the help of the notion of triangle−free graphs and Mantel's theorem. Triangle−free graphs have a maximum number of edges that reveals their characteristics and practical applications.
Many fields, including network analysis, social network modelling, and algorithm creation, can benefit from this finding.Mantel's theorem allows us to examine network connection, optimise graph algorithms, and discover novel graph architectures. The theorem also provides a springboard for further explorations into the characteristics and interrelationships of graphs, paving the way for future study and development in the field of graph theory.
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