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 A3, then the number of triangle in Undirected Graph is equal to trace(A3) / 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 + A + .... + A[V-1][V-1] Count of triangles = trace(A3) / 6
Below is implementation of above formula.
Total number of Triangle in Graph : 2
How does this work?
If we compute An for an adjacency matrix representation of graph, then a value An[i][j] represents number of distinct walks between vertex i to j in graph. In A3, 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 A3[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(A3) / 6.
The time complexity of above algorithm is O(V3) where V is number of vertices in the graph, we can improve the performance to O(V2.8074) using Strassen’s matrix multiplication algorithm.
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.
- Number of Triangles in Directed and Undirected Graphs
- Count number of edges in an undirected graph
- Number of single cycle components in an undirected graph
- Undirected graph splitting and its application for number pairs
- Program to count Number of connected components in an undirected graph
- Maximum number of edges among all connected components of an undirected graph
- Convert the undirected graph into directed graph such that there is no path of length greater than 1
- Clone an Undirected Graph
- Detect cycle in an undirected graph using BFS
- Sum of degrees of all nodes of a undirected graph
- Print all the cycles in an undirected graph
- Connected Components in an undirected graph
- Eulerian Path in undirected graph
- Detect cycle in an undirected graph
- Find k-cores of an undirected graph