High-diversity species assemblages are very common in nature, and yet the factors allowing for the maintenance of biodiversity remain obscure. The competitive exclusion principle and May's complexity-diversity puzzle both suggest that a community can support only a small number of species, turning the spotlight on the dynamics of local patches or islands, where stable and uninvadable (SU) subse...

Albertson conjectured that if graph G has chromatic number r, then the crossing number of G is at least that of the complete graph Kr. This conjecture in the case r = 5 is equivalent to the four color theorem. It was verified for r = 6 by Oporowski and Zhao. In this paper, we prove the conjecture for 7 ≤ r ≤ 12 using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on...

Clubs are generalizations of cliques. For a positive integer s, an s-club in a graph G is a set of vertices that induces a subgraph of G of diameter at most s. The importance and fame of cliques are evident, whereas clubs provide more realistic models for practical applications. Computing an s-club of maximum cardinality is an NP-hard problem for every fixed s ≥ 1, and this problem has attracte...

The vertex set of a halved cube Qd consists of a bipartition vertex set of a cube Qd and two vertices are adjacent if they have a common neighbour in the cube. Let d ≥ 5. Then it is proved that Qd is the only connected, ( d 2 ) -regular graph on 2d−1 vertices in which every edge lies in two d-cliques and two d-cliques do not intersect in a vertex.

We investigate a simple stochastic model of social network formation by the process of reinforcement learning with discounting of the past. In the limit, for any value of the discounting parameter, small, stable cliques are formed. However, the time it takes to reach the limiting state in which cliques have formed is very sensitive to the discounting parameter. Depending on this value, the limi...

In the Correlation Clustering problem, also known as Cluster Editing, we are given an undirected graph G and a positive integer k; the task is to decide whether G can be transformed into a cluster graph, i.e., a disjoint union of cliques, by changing at most k adjacencies, that is, by adding or deleting at most k edges. The motivation of the problem stems from various tasks in computational bio...

We explore neural network and related heuristic methods for the fast approximate solution of the Maximum Clique problem. One of these algorithms, Mean Field Annealing, is implemented on the Connection Machine CM-5 and a fast annealing schedule is experimentally evaluated on random graphs, as well as on several benchmark graphs. The other algorithms, which perform certain randomized local search...

In the hidden clique problem, one needs to find the maximum clique in an n-vertex graph that has a clique of size k but is otherwise random. An algorithm of Alon, Krivelevich and Sudakov that is based on spectral techniques is known to solve this problem (with high probability over the random choice of input graph) when k ≥ c √ n for a sufficiently large constant c. In this manuscript we presen...

Identifying clusters of similar objects in data plays a significant role in a wide range of applications. As a model problem for clustering, we consider the densest k-disjoint-clique problem, whose goal is to identify the collection of k disjoint cliques of a given weighted complete graph maximizing the sum of the densities of the complete subgraphs induced by these cliques. In this paper, we e...

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,...