Given an Undirected simple graph, We need to find how many triangles it can have. For example below graph have 2 triangles in it.

Let A[][] be adjacency matrix representation of graph. If we calculate A^{3}, then the number of triangle in Undirected Graph is equal to trace(A^{3}) / 6. Where trace(A) is the sum of the elements on the main diagonal of matrix A.

Trace of a graph represented as adjacency matrix A[V][V] is, trace(A[V][V]) = A[0][0] + A[1][1] + .... + A[V-1][V-1] Count of triangles = trace(A^{3}) / 6

Below is C++ implementation of above formula.

// A C++ program for finding number of triangles in an // Undirected Graph. The program is for adjacency matrix // representation of the graph #include <bits/stdc++.h> using namespace std; // Number of vertices in the graph #define V 4 // Utility function for matrix multiplication void multiply(int A[][V], int B[][V], int C[][V]) { for (int i = 0; i < V; i++) { for (int j = 0; j < V; j++) { C[i][j] = 0; for (int k = 0; k < V; k++) C[i][j] += A[i][k]*B[k][j]; } } } // Utility function to calculate trace of a matrix (sum of // diagnonal elements) int getTrace(int graph[][V]) { int trace = 0; for (int i = 0; i < V; i++) trace += graph[i][i]; return trace; } // Utility function for calculating number of triangles in graph int triangleInGraph(int graph[][V]) { int aux2[V][V]; // To Store graph^2 int aux3[V][V]; // To Store graph^3 // Initialising aux matrices with 0 for (int i = 0; i < V; ++i) for (int j = 0; j < V; ++j) aux2[i][j] = aux3[i][j] = 0; // aux2 is graph^2 now printMatrix(aux2); multiply(graph, graph, aux2); // after this multiplication aux3 is // graph^3 printMatrix(aux3); multiply(graph, aux2, aux3); int trace = getTrace(aux3); return trace / 6; } // driver program to test above function int main() { /* Let us create the example graph discussed above */ int graph[V][V] = {{0, 1, 1, 0}, {1, 0, 1, 1}, {1, 1, 0, 1}, {0, 1, 1, 0} }; printf("Total number of Triangle in Graph : %d\n", triangleInGraph(graph)); return 0; }

Output:

Total number of Triangle in Graph : 2

**How does this work?**

If we compute A^{n} for an adjacency matrix representation of graph, then a value A^{n}[i][j] represents number of distinct walks between vertex i to j in graph. In A^{3}, we get all distinct paths of length 3 between every pair of vertices.

A triangle is a cyclic path of length three, i.e. begins and ends at same vertex. So A^{3}[i][i] represents a triangle beginning and ending with vertex i. Since a triangle has three vertices and it is counted for every vertex, we need to divide result by 3. Furthermore, since the graph is undirected, every triangle twice as i-p-q-j and i-q-p-j, so we divide by 2 also. Therefore, number of triangles is trace(A^{3}) / 6.

**Time Complexity: **

The time complexity of above algorithm is O(V^{3}) where V is number of vertices in the graph, we can improve the performance to O(V^{2.8074}) using Strassen’s matrix multiplication algorithm.

**References:**

http://www.d.umn.edu/math/Technical%20Reports/Technical%20Reports%202007-/TR%202012/yang.pdf

Number of Triangles in Directed and Undirected Graphs

This article is contributed by Utkarsh Trivedi. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.