1.1 Maximum Clique Problem Assume that the finite undirected simple graph is given, where V is the set of nodes, E is the set of edges. The arbitrary full graph is called a clique. The clique, which does not contain other cliques, is called a maximal Clique. The largest maximal clique is called a maximum clique. To extract all maximal cliques from the graph G. Many algorithms have...

In this note, we show that a complete k-partite graph is the only graph with clique number k among all degree-equivalent simple graphs. This result gives a lower bound on the clique number, which is sharper than existing bounds on a large family of graphs.

We show that for a graph G it is NP-hard to decide whether its independence number (G) equals its clique partition number (G) even when some minimum clique partition of G is given. This implies that any (G)-upper bound provably better than (G) is NPhard to compute. To establish this result we use a reduction of the quasigroup completion problem (QCP, known to be NP-complete) to the maximum inde...

This paper examines methods for predicting and estimating the number of maximal cliques in a random graph. A clique is a subgraph where each vertex is connected to every other vertex in the subgraph. A maximal clique is a clique which is not a proper subgraph of another clique. There are many algorithms that enumerate all maximal cliques in a graph, but since the task can take exponential time,...

This document details the characteristics, input parameters and suggested usage of the three graph generators included in this suite. 1 SSCA#2 graph generator This generator produces graphs used in the DARPA HPCS SSCA#2 benchmark [1]. The SSCA#2 graph is directed with integer edge weights, and made up of random-sized cliques, with a hierarchical inter-clique distribution of edges based on a dis...

In this paper, we analyze the evolution of a small-world network and its subsequent transformation to a random network using the idea of link rewiring under the well-known Watts-Strogatz model for complex networks. Every link u-v in the regular network is considered for rewiring with a certain probability and if chosen for rewiring, the link u-v is removed from the network and the node u is con...

Because of unpredictable, uncertain and time-varying nature of real networks it seems that stochastic graphs, in which weights associated to the edges are random variables, may be a better candidate as a graph model for real world networks. Once the graph model is chosen to be a stochastic graph, every feature of the graph such as path, clique, spanning tree and dominating set, to mention a few...

In this paper the performance of four well-known metaheuristics consisting of Artificial Bee Colony (ABC), Biogeographic Based Optimization (BBO), Harmony Search (HS) and Teaching Learning Based Optimization (TLBO) are investigated on optimal domain decomposition for parallel computing. A clique graph is used for transforming the connectivity of a finite element model (FEM) into that of the cor...