In graph idea, a clique is a subset of vertices of an undirected graph such that each two distinct vertices within the clique are adjoining, that’s, they’re linked by an fringe of the graph. The variety of cliques in a graph is the entire variety of cliques that may be discovered within the graph.
The Arithmetic behind cliques in a graph entails the idea of adjacency matrices and graph idea. An adjacency matrix is a matrix illustration of a graph the place every row and column corresponds to a vertex within the graph and the weather of the matrix point out whether or not there’s an edge between the vertices.Â
For instance, in an undirected graph with 5 vertices, the adjacency matrix can be a 5Ã—5 matrix the place a 1 in place (i, j) signifies that there’s an edge between vertex i and vertex j, and a 0 signifies that there isn’t any edge.
1 1 1 0 0
1 1 1 0 0
1 1 1 1 1
0 0 1 1 1
0 0 1 1 1
To search out the variety of cliques in a graph utilizing an adjacency matrix, you should utilize a graph algorithm such because the Bronâ€“Kerbosch algorithm, which is an environment friendly methodology for enumerating all cliques in an undirected graph.
The Bronâ€“Kerbosch algorithm works by iterating over all potential subsets of the vertices and checking in the event that they kind a clique. It does this through the use of the adjacency matrix to test if each pair of vertices within the subset is adjoining (i.e., linked by an edge). If they’re, then the subset varieties a clique and is added to the listing of cliques.
Instance 1:
Take into account the next graph:
This graph has three cliques: {1, 2, 3}, {3, 4, 5}, and {1, 2, 4, 5}.
To search out the variety of cliques in a graph, graph traversal algorithms corresponding to depthfirst search (DFS) or breadthfirst search (BFS) can be utilized to go to all of the vertices and test for cliques at every vertex.
For instance, you could possibly begin at vertex 1 and carry out a DFS to discover the graph. If you attain vertex 3, you already know that the vertices 1, 2, and three kind a clique. You possibly can then proceed the DFS to discover the remaining vertices and test for cliques at every vertex.
Alternatively, you could possibly use a graph algorithm particularly designed to seek out cliques, such because the Bronâ€“Kerbosch algorithm. This algorithm is an environment friendly methodology for enumerating all cliques in an undirected graph.
Instance 2:
Take into account a easy undirected graph with 4 vertices and 6 edges, as proven under:
To depend the variety of cliques on this graph, we will use the next method:
Variety of cliques = n * (n â€“ 1) / 2 â€“ m + 1 the place n is the variety of vertices within the graph and m is the variety of edges. Plugging within the values for this graph, we get:
Variety of cliques = 4 * (4 â€“ 1) / 2 â€“ 6 + 1 = 2So there are 2 cliques on this graph.
This method works as a result of it counts the variety of potential pairs of vertices within the graph (n * (n â€“ 1) / 2), after which subtracts the variety of edges to account for overcounting. Lastly, it provides 1 to account for the truth that a single vertex by itself can be thought of a clique.
This method solely works for undirected graphs, and it might not give the proper outcome for graphs with a number of edges or selfloops. Additionally it is not sensible for big graphs, because the time complexity of this method is O(n^2). Nevertheless, it may be a great tool for rapidly counting the variety of cliques in small graphs
Approaches:
 Bruteforce search: One method is to easily enumerate all potential cliques within the graph and depend them. This method has a time complexity of O(3^n), the place n is the variety of vertices within the graph, so it’s only sensible for very small graphs.
 BronKerbosch algorithm: The BronKerbosch algorithm is a pivotbased algorithm that makes use of a recursive method to seek out all cliques in a graph. It has a time complexity of O(3^(n/3)), the place n is the variety of vertices within the graph, and an area complexity of O(n).
 Tomita algorithm: The Tomita algorithm is one other pivotbased algorithm that makes use of a recursive method to seek out all cliques in a graph. It has a time complexity of O(4^(n/4)), the place n is the variety of vertices within the graph and an area complexity of O(n).
 Pivot BronKerbosch algorithm: The pivot BronKerbosch algorithm is an enchancment over the BronKerbosch algorithm that makes use of a pivot aspect to prune the search area and scale back the time complexity. It has a time complexity of O(2^n), the place n is the variety of vertices within the graph, and an area complexity of O(n^2).
 Hybrid algorithm: The hybrid algorithm is a mixture of the BronKerbosch and Tomita algorithms that makes use of each pivoting and recursion to seek out all cliques in a graph. It has a time complexity of O(2^n), the place n is the variety of vertices within the graph and an area complexity of O(n^2)
 Approximation algorithms: One other method is to make use of approximation algorithms, that are designed to discover a good approximation of the variety of cliques in a graph in polynomial time. These algorithms might not discover all cliques within the graph, however they are often helpful in instances the place the precise variety of cliques will not be obligatory.
 Parallelization methods: Additionally it is potential to enhance the efficiency of the above algorithms through the use of parallelization methods, corresponding to multithreading or distributed computing, to divide the work throughout a number of processors. Nevertheless, these approaches might have extra overhead prices and will not be appropriate for every type of graphs.
Which method is greatest to your particular drawback will rely upon the dimensions of the graph and the specified time and area complexity. It’s best to select the method that greatest meets your wants and constraints.
Remark behind the Approaches:
The completely different approaches to discovering the variety of cliques in a graph are primarily based on completely different observations and methods. Listed below are some observations behind these approaches:
 Bruteforce search: This method merely enumerates all potential cliques within the graph and counts them. It’s primarily based on the commentary {that a} clique is a subset of the vertices of the graph which might be all linked to one another.
 BronKerbosch algorithm: The BronKerbosch algorithm relies on the commentary {that a} clique could be discovered by beginning with a vertex after which increasing to incorporate its neighbors, so long as they’re all linked to one another. The algorithm makes use of a pivot aspect to prune the search area and scale back the time complexity.
 Tomita algorithm: The Tomita algorithm relies on the commentary {that a} clique could be discovered by beginning with a vertex after which increasing to incorporate its neighbors, so long as they’re all linked to one another. The algorithm makes use of a pivot aspect to prune the search area and scale back the time complexity.
 Pivot BronKerbosch algorithm: The pivot BronKerbosch algorithm is an enchancment over the BronKerbosch algorithm that makes use of a pivot aspect to prune the search area and scale back the time complexity. It’s primarily based on the commentary that many cliques in a graph share widespread vertices, and through the use of a pivot aspect to focus the search on these vertices, it’s potential to scale back the time complexity of the algorithm.
 Hybrid algorithm: The hybrid algorithm combines the BronKerbosch and Tomita algorithms and makes use of each pivoting and recursion to seek out all cliques in a graph. It’s primarily based on the commentary that each of those algorithms are efficient at discovering cliques, and by combining them, it’s potential to seek out all cliques in a graph extra effectively.
 Approximation algorithms: Approximation algorithms are primarily based on the commentary that it’s usually ample to discover a good approximation of the variety of cliques in a graph, quite than the precise quantity. These algorithms use methods corresponding to random sampling or graph partitioning to discover a good approximation in polynomial time.
 Parallelization methods: Parallelization methods are primarily based on the commentary that it’s usually potential to enhance the efficiency of an algorithm by dividing the work throughout a number of processors. These methods can be utilized to hurry up the execution of the above algorithms by dividing the search area amongst a number of threads or processes
Right here is a straightforward algorithm in Python to seek out the variety of cliques in an undirected graph:
Python3

Variety of cliques: 2 Cliques: [{'D', 'B', 'C', 'A'}, {'E', 'F'}]
The time complexity of the find_cliques operate supplied within the code is O(n+m), the place n is the variety of vertices within the graph and m is the variety of edges. It is because the operate performs a depthfirst search (DFS) on the graph, which takes time proportional to the variety of vertices and
The auxiliary area of this operate is O(n) as a result of it makes use of a set to retailer the visited vertices and a listing to retailer the cliques, each of which have sizes proportional to the variety of vertices within the graph.
Be aware: that the time and area complexity of this operate could also be completely different if the graph is represented otherwise, for instance as an adjacency matrix as an alternative of an adjacency listing. The time and area complexity of an algorithm can even rely upon the precise implementation and the precise enter.
Time and area complexity of every method:
Strategy  Time Complexity  House Complexity 

BronKerbosch algorithm 
O(3^(n/3)) 
Â O(n) 
Tomita algorithm 
O(4^(n/4)) 
O(n) 
Pivot BronKerbosch algorithm 
O(2^n) 
Â O(2^n) 
Hybrid algorithm 
O(2^n) 
Â O(2^n) 
 The BronKerbosch algorithm and the Tomita algorithm are each pivotbased algorithms that use a recursive method to seek out all cliques in a graph.Â
 The Pivot BronKerbosch algorithm is an enchancment over the BronKerbosch algorithm that reduces the time complexity through the use of a pivot aspect to prune the search area.Â
 The hybrid algorithm is a mixture of the BronKerbosch and Tomita algorithms that makes use of each pivoting and recursion to seek out all cliques in a graph.
 It’s price noting that the time complexity of those algorithms could be improved through the use of parallelization methods or approximative algorithms, however these approaches might not discover all cliques within the graph and will have extra overhead prices.