Given a Graph, count number of triangles in it. The graph is can be directed or undirected.

Example:

Input: digraph[V][V] = { {0, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 0, 0}, {0, 0, 1, 0} }; Output: 2 Give adjacency matrix represents following directed graph.

We have discussed a method based on graph trace that works for undirected graphs. In this post a new method is discussed with that is simpler and works for both directed and undirected graphs.

The idea is to use three nested loops to consider every triplet (i, j, k) and check for the above condition (there is an edge from i to j, j to k and k to i)

However in an **undirected graph**, the triplet (i, j, k) can be permuted to give six combination (See previous post for details). Hence we divide the total count by 6 to get the actual number of triangles.

In case of **directed graph**, the number of permutation would be 3 (as order of nodes becomes relevant). Hence in this case the total number of triangles will be obtained by dividing total count by 3. For example consider the directed graph given below

Following is the implementation.

## C/C++

// C++ program to count triangles // in a graph. The program is for // adjacency matrix representation // of the graph. #include<bits/stdc++.h> // Number of vertices in the graph #define V 4 using namespace std; // function to calculate the // number of triangles in a // simple directed/undirected // graph. isDirected is true if // the graph is directed, its // false otherwise int countTriangle(int graph[V][V], bool isDirected) { // Initialize result int count_Triangle = 0; // Consider every possible // triplet of edges in graph for (int i = 0; i < V; i++) { for (int j = 0; j < V; j++) { for (int k = 0; k < V; k++) { // check the triplet if // it satisfies the condition if (graph[i][j] && graph[j][k] && graph[k][i]) count_Triangle++; } } } // if graph is directed , // division is done by 3, // else division by 6 is done isDirected? count_Triangle /= 3 : count_Triangle /= 6; return count_Triangle; } //driver function to check the program int main() { // Create adjacency matrix // of an undirected graph int graph[][V] = { {0, 1, 1, 0}, {1, 0, 1, 1}, {1, 1, 0, 1}, {0, 1, 1, 0} }; // Create adjacency matrix // of a directed graph int digraph[][V] = { {0, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 0, 0}, {0, 0, 1, 0} }; cout << "The Number of triangles in undirected graph : " << countTriangle(graph, false); cout << "\n\nThe Number of triangles in directed graph : " << countTriangle(digraph, true); return 0; }

## Java

// Java program to count triangles // in a graph. The program is // for adjacency matrix // representation of the graph. import java.io.*; class Triangles { // Number of vertices in the graph int V = 4; // function to calculate the number // of triangles in a simple // directed/undirected graph. isDirected // is true if the graph is directed, // its false otherwise. int countTriangle(int graph[][], boolean isDirected) { // Initialize result int count_Triangle = 0; // Consider every possible // triplet of edges in graph for (int i = 0; i < V; i++) { for (int j = 0; j < V; j++) { for (int k=0; k<V; k++) { // check the triplet if it // satisfies the condition if (graph[i][j] == 1 && graph[j][k] == 1 && graph[k][i] == 1) count_Triangle++; } } } // if graph is directed , division // is done by 3 else division // by 6 is done if(isDirected == true) { count_Triangle /= 3; } else { count_Triangle /= 6; } return count_Triangle; } // driver code public static void main(String args[]) { // Create adjacency matrix // of an undirected graph int graph[][] = {{0, 1, 1, 0}, {1, 0, 1, 1}, {1, 1, 0, 1}, {0, 1, 1, 0} }; // Create adjacency matrix // of a directed graph int digraph[][] = { {0, 0, 1, 0}, {1, 0, 0, 1}, {0, 1, 0, 0}, {0, 0, 1, 0} }; System.out.println("The Number of triangles "+ "in undirected graph : " + countTriangle(graph, false)); System.out.println("\n\nThe Number of triangles"+ " in directed graph : "+ countTriangle(digraph, true)); } } // This code is contributed by Anshika Goyal.

## Python

# Python program to count triangles in a graph. The program is # for adjacency matrix representation of the graph. # function to calculate the number of triangles in a simple # directed/undirected graph. # isDirected is true if the graph is directed, its false otherwise def countTriangle(g, isDirected): nodes = len(g) count_Triangle = 0 #Initialize result # Consider every possible triplet of edges in graph for i in range(nodes): for j in range(nodes): for k in range(nodes): # check the triplet if it satisfies the condition if( i!=j and i !=k and j !=k and g[i][j] and g[j][k] and g[k][i]): count_Triangle += 1 # if graph is directed , division is done by 3 # else division by 6 is done return count_Triangle/3 if isDirected else count_Triangle/6 # Create adjacency matrix of an undirected graph graph = [[0, 1, 1, 0], [1, 0, 1, 1], [1, 1, 0, 1], [0, 1, 1, 0 ]] # Create adjacency matrix of a directed graph digraph = [[0, 0, 1, 0], [1, 0, 0, 1], [0, 1, 0, 0], [0, 0, 1, 0 ]] print ("The Number of triangles in undirected graph : %d" %countTriangle(graph, False)) print ("The Number of triangles in directed graph : %d" %countTriangle(digraph, True)) # This code is contributed by Neelam Yadav

Output:

The Number of triangles in undirected graph : 2 The Number of triangles in directed graph : 2

**Comparison of this approach with previous approach:**

Advantages:

- No need to calculate Trace.
- Matrix- multiplication is not required.
- Auxiliary matrices are not required hence optimized in space.
- Works for directed graphs.

Disadvantages:

- The time complexity is O(n
^{3}) and can’t be reduced any further.

This article is contributed by **Ashutosh Kumar**. If you like GeeksforGeeks and would like to contribute, you can also write an article and mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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