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In this paper, we relate the problem of finding a maximum clique to the intersection number of the input graph (i.e. the minimum number of cliques needed to edge cover the graph). In particular, we consider the maximum clique problem for…

In this paper we prove the strong consistency of several methods based on the spectral clustering techniques that are widely used to study the community detection problem in stochastic block models (SBMs). We show that under some weak…

Two of the most fundamental distributed symmetry-breaking problems are that of finding a maximal independent set (MIS) and a maximal matching (MM) in a graph. It is a major open question whether these problems can be solved in constant…

Erd\H{o}s and Moser raised the question of determining the maximum number of maximal cliques or equivalently, the maximum number of maximal independent sets in a graph on $n$ vertices. Since then there has been a lot of research along these…

The maximum clique problem is a classical NP-complete problem in graph theory and has important applications in many domains. In this paper we show, in a partially non-constructive way, the existence of an exact polynomial-time algorithm…

Finding cohesive subgraphs in a large graph has many important applications, such as community detection and biological network analysis. Clique is often a too strict cohesive structure since communities or biological modules rarely form as…

The densest $k$-subgraph problem is the problem of finding a $k$-vertex subgraph of a graph with the maximum number of edges. In order to solve large instances of the densest $k$-subgraph problem, we introduce two algorithms that are based…

Point sets matching method is very important in computer vision, feature extraction, fingerprint matching, motion estimation and so on. This paper proposes a robust point sets matching method. We present an iterative algorithm that is…

Given a collection of $m$ sets from a universe $\mathcal{U}$, the Maximum Set Coverage problem consists of finding $k$ sets whose union has largest cardinality. This problem is NP-Hard, but the solution can be approximated by a polynomial…

A disk graph is an intersection graph of disks in $\mathbb{R}^2$. Determining the computational complexity of finding a maximum clique in a disk graph is a long-standing open problem. In 1990, Clark, Colbourn, and Johnson gave a…

One of the most useful measures of cluster quality is the modularity of a partition, which measures the difference between the number of the edges joining vertices from the same cluster and the expected number of such edges in a random…

This paper develops a flexible distribution-free method for collective outlier detection and enumeration, designed for situations in which the presence of outliers can be detected powerfully even though their precise identification may be…

This article explores and analyzes the unsupervised clustering of large partially observed graphs. We propose a scalable and provable randomized framework for clustering graphs generated from the stochastic block model. The clustering is…

We asymptotically determine the maximum density of subgraphs isomorphic to $H$, where $H$ is any graph containing a dominating vertex, in graphs $G$ on $n$ vertices with bounded maximum degree and bounded clique number. That is, we…

This work concerns sampling of smooth signals on arbitrary graphs. We first study a structured sampling strategy for such smooth graph signals that consists of a random selection of few pre-defined groups of nodes. The number of groups to…

A $k$-defective clique is a relaxation of the traditional clique definition, allowing up to $k$ missing edges. This relaxation is crucial in various real-world applications such as link prediction, community detection, and social network…

We consider the fundamental problem of detecting/counting copies of a fixed pattern graph in a host graph. The recent progress on this problem has not included complete pattern graphs, i.e., cliques (and their complements, i.e., edge-free…

We study finding and listing $k$-cliques in a graph, for constant $k\geq 3$, a fundamental problem of both theoretical and practical importance. Our main contribution is a new output-sensitive algorithm for listing $k$-cliques in graphs,…

A graph is inductive $k$-independent if there exists and ordering of its vertices $v_{1},...,v_{n}$ such that $\alpha(G[N(v_{i})\cap V_{i}])\leq k $ where $N(v_{i})$ is the neighborhood of $v_{i}$, $V_{i}=\{v_{i},...,v_{n}\}$ and $\alpha$…

Since its first use by Euler on the problem of the seven bridges of K\"onigsberg, graph theory has shown excellent abilities in solving and unveiling the properties of multiple discrete optimization problems. The study of the structure of…