Finding Cliques of a Graph using Prime Numbers

A clique in an undirected graph G= (V, E) is a subset V' V of vertices, each pair of which is connected by an edge in E. The clique problem is an optimization problem of finding a clique of maximum size in graph. The clique problem is NP-Complete. We have succeeded in developing a fast algorithm for maximum clique problem by employing the method of verification and elimination. For a graph of size N there are 2N sub graphs, which may be cliques and hence verifying all of them...

We investigate a number of recently reported exact algorithms for the maximum clique problem (MCQ, MCR, MCS, BBMC). The program code used is presented and critiqued showing how small changes in implementation can have a drastic effect on performance. The computational study demonstrates how problem features and hardware platforms influence algorithm behaviour. The minimum width order (smallest-last) is investigated, and MCS is broken into its consituent parts and we discover ...

In this paper we continue our investgation of the prime graph of a finite simple group started in http://arxiv.org/abs/math/0506294 (the printed version appeared in [1]). We describe all cocliques of maximal size for all finite simple groups and also we correct mistakes and misprints from our previous paper. The list of correction is given in Appendix of the present paper.

We propose a fast, parallel maximum clique algorithm for large sparse graphs that is designed to exploit characteristics of social and information networks. The method exhibits a roughly linear runtime scaling over real-world networks ranging from 1000 to 100 million nodes. In a test on a social network with 1.8 billion edges, the algorithm finds the largest clique in about 20 minutes. Our method employs a branch and bound strategy with novel and aggressive pruning techniques...

This manuscript provides a comprehensive review of the Maximum Clique Problem, a computational problem that involves finding subsets of vertices in a graph that are all pairwise adjacent to each other. The manuscript covers in a simple way classical algorithms for solving the problem and includes a review of recent developments in graph neural networks and quantum algorithms. The review concludes with benchmarks for testing classical as well as new learning, and quantum algor...

A simple graph on $n$ vertices may contain a lot of maximum cliques. But how many can it potentially contain? We will define prime and composite graphs, and we will show that if $n \ge 15$, then the grpahs with the maximum number of maximum cliques have to be composite. Moreover, we will show an edge bound from which we will prove that if any factor of a composite graph has $\omega(G_i) \ge 5$, then it cannot have the maximum number of maximum cliques. Using this we will show...

We take an existing implementation of an algorithm for the maximum clique problem and modify it so that we can distribute it over an ad-hoc cluster of machines. Our goal was to achieve a significant speedup in performance with minimal development effort, i.e. a maximum costup. We present a simple modification to a state-of-the-art exact algorithm for maximum clique that allows us to distribute it across many machines. An empirical study over large hard benchmarks shows that s...

Let PR$[n]$ be the graph whose vertices are $2,3,\ldots,n$ with vertex $v$ adjacent to vertex $w$ if and only if $\gcd(v,w)>1$. It is shown that $\pi(n)$, the the number of primes no more than $n$, equals the Lov\'{a}sz number of this graph. This result suggests new avenues for graph-theoretic investigations of number-theoretic problems.

The decomposition of complex networks into smaller, interconnected components is a central challenge in network theory with a wide range of potential applications. In this paper, we utilize tools from group theory and ring theory to study this problem when the network is a Cayley graph. In particular, we answer the following question: Which Cayley graphs are prime?

The MaxClique problem, finding the largest complete subgraph in an Erd{\"o}s-R{\'e}nyi $G(N,p)$ random graph in the large $N$ limit, is a well-known example of a simple problem for which finding any approximate solution within a factor of $2$ of the known, probabilistically determined limit, appears to require P$=$NP. This type of search has practical importance in very large graphs. Algorithmic approaches run into phase boundaries long before they reach the size of the large...