Cliques In Graphs Research Articles

Cliques in realization graphs

The realization graph G(d) of a degree sequence d is the graph whose vertices are labeled realizations of d, where edges join realizations that differ by swapping a single pair of edges. Barrus (2016) [3] characterized d for which G(d) is triangle-free. Here, for any n≥4, we describe a structure in realizations of d that exactly determines whether G(d) has a clique of size n. As a consequence we determine the degree sequences d for which G(d) is a complete graph on n vertices.

Properties and Recognition of Atom Graphs

The atom graph of a connected graph is a graph whose vertices are the atoms obtained by clique minimal separator decomposition of this graph, and whose edges are the edges of all its atom trees. A graph G is an atom graph if there is a graph whose atom graph is isomorphic to G. We study the class of atom graphs, which is also the class of atom graphs of chordal graphs, and the associated recognition problem. We prove that each atom graph is a perfect graph and give a characterization of atom graphs in terms of a spanning tree, inspired by the characterization of clique graphs of chordal graphs as expanded trees. We also characterize the chordal graphs having the same atom and clique graph, and solve the recognition problem of atom graphs of two graph classes.

Van Lint–MacWilliams' conjecture and maximum cliques in Cayley graphs over finite fields

A well-known conjecture due to van Lint and MacWilliams states that if A is a subset of Fq2 such that 0,1∈A, |A|=q, and a−b is a square for each a,b∈A, then A must be the subfield Fq. This conjecture is often phrased in terms of the maximum cliques in Paley graphs. It was first proved by Blokhuis and later extended by Sziklai to generalized Paley graphs. In this paper, we give a new proof of the conjecture and its variants, and show how this Erdős-Ko-Rado property of Paley graphs extends to a larger family of Cayley graphs, which we call Peisert-type graphs. These results resolve the conjectures by Mullin and Yip.

Random cliques in random graphs and sharp thresholds for F$$ F $$‐factors

AbstractWe show that for each , in a density range extending up to, and slightly beyond, the threshold for a ‐factor, the copies of in the random graph are randomly distributed, in the (one‐sided) sense that the hypergraph that they form contains a copy of a binomial random hypergraph with almost exactly the right density. Thus Jeff Kahn's recent asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem implies a corresponding bound for the threshold for to contain a ‐factor. The case is more difficult, and has been settled by Annika Heckel. We also prove a corresponding result for ‐factors in random ‐uniform hypergraphs, as well as (in some cases weaker) generalizations replacing by certain other (hyper)graphs.

On the maxima of motzkin-straus programs and cliques of graphs

On the maxima of motzkin-straus programs and cliques of graphs

Cellular goore game with application to finding maximum clique in social networks

Abstract The goore game (GG) is a model for collective decision making under uncertainty, which can be used as a tool for stochastic optimization of a discrete variable function. The GG has a fascinating property that can be resolved in a distributed manner with no intercommunication between the players. The game has found applications in many network applications, including sensor networks, quality-of-service routing, and social networks. In this paper, we introduce an extension of GG called cellular goore game (CGG) for the first time. The CGG is a network of GGs. In this network, each node (or subset of nodes in the network) plays the rule of referees, each of which participates in a GG with its neighboring players (voters) at any time. Like in GG, each player independently selects its optimal action between two available actions based on their gains and losses received from its adjacent referee. Players in CGG know nothing about how other players are playing or even how/why they are rewarded/penalized. The potential of the CGG is shown by providing an algorithm for finding a maximum clique in social networks. Our research provides the first-time study of the CGG for finding a maximum clique in graphs. The performance of the CGG-based algorithm for finding maximum clique is studied on the standard clique benchmark called DIMACS by several experiments. The obtained result shows that the CGG-based algorithm is superior to the existing algorithms in terms of finding maximum clique size and time.

On the clique behavior of graphs of low degree

To any simple graph \(G\), the clique graph operator \(K\) associates the graph \(K(G)\) which is the intersection graph of the maximal complete subgraphs of \(G\). The iterated clique graphs are defined by \(K^{0}(G)=G\) and \(K^{n}(G)=K(K^{n-1}(G))\) for \(n\ge 1\). If there are \(m<n\) such that \(K^{m}(G)\) is isomorphic to \(K^{n}(G)\) we say that \(G\) is convergent, otherwise, \(G\) is divergent. The first example of a divergent graph was shown by Neumann-Lara in the 1970s, and is the graph of the octahedron. In this paper, we prove that among the connected graphs with maximum degree 4, the octahedron is the only one that is divergent.

Computing a maximum clique in geometric superclasses of disk graphs

In the 90’s Clark, Colbourn and Johnson wrote a seminal paper where they proved that maximum clique can be solved in polynomial time in unit disk graphs. Since then, the complexity of maximum clique in intersection graphs of d-dimensional (unit) balls has been investigated. For ball graphs, the problem is NP-hard, as shown by Bonamy et al. (FOCS ’18). They also gave an efficient polynomial time approximation scheme (EPTAS) for disk graphs. However, the complexity of maximum clique in this setting remains unknown. In this paper, we show the existence of a polynomial time algorithm for a geometric superclass of unit disk graphs. Moreover, we give partial results toward obtaining an EPTAS for intersection graphs of convex pseudo-disks.

Clique dynamics of locally cyclic graphs with δ ≥ 6

We prove that the clique graph operator k is divergent on a locally cyclic graph G (i.e. NG(v) is a circle) with minimum degree δ(G)=6 if and only if G is 6-regular. The clique graph kG of a graph G has the maximal complete subgraphs of G as vertices, and the edges are given by non-empty intersections. If all iterated clique graphs of G are pairwise non-isomorphic, the graph G is k-divergent; otherwise, it is k-convergent.To prove our claim, we explicitly construct the iterated clique graphs of those infinite locally cyclic graphs with δ≥6 which induce simply connected simplicial surfaces. These graphs are k-convergent if the size of triangular-shaped subgraphs of a specific type is bounded from above. We apply this criterion by using the universal cover of the triangular complex of an arbitrary finite locally cyclic graph with δ=6, which shows our divergence characterisation.

On a correspondence between maximal cliques in Paley graphs of square order

Let q be an odd prime power. Denote by r(q) the value of q modulo 4. In this paper, we establish a linear fractional correspondence between two types of maximal cliques of size q+r(q)2 in the Paley graph of order q2.

Dynamic hypergraphs of renewal processes in mobile networks.

The characteristics of random geometric hypergraphs are studied as mathematical models of scalable wireless computer networks. An efficient algorithm for finding cliques in geometric graphs, constructing hypergraphs from geometric configurations has been developed. The types of hyper-edges in hypergraphs generated by a scalable configuration have been identified. The influence of random failures of nodes of computer networks and their restorations on the dynamics of hypergraphs of networks is considered. The analysis of the dynamics of the number of active nodes depending on the type of probability distributions of uptime and recovery time is carried out. The dependences of the mathematical expectation of the number of hyper-edges of certain types in the geometric hypergraph of a wireless computer network on the network operation time, on the radii of zones of reliable reception / transmission of a signal, on the ratio of the parameters of local recovery processes are obtained. The presentation of the results is accompanied by charts.

On fast enumeration of maximal cliques in large graphs

Maximal Clique Enumeration (MCE) is a fundamental and challenging problem in graph theory and various network applications. Numerous algorithms have been proposed in the past decades, however, only a few of them focus on improving the practical efficiency in large graphs. To this end, we propose an efficient algorithm called FACEN based on the Bron–Kerbosch framework. To optimize the memory and time consumption, we apply a hybrid data structure with adjacency list and partial adjacency matrix, and introduce a dynamic pivot selection rule based on the degeneracy order. FACEN is evaluated on a total of 64 benchmark instances from various sources. Computational results indicate that the proposed algorithm is highly competitive with the current leading MCE methods. In particular, our algorithm is able to enumerate all maximal cliques on the tested real-world social networks with millions of vertices and edges. For very large graphs, we provide an additional experiment for solving the MCE variant with lower bound, and investigate the benefits of FACEN.

Finding Maximum k-Cliques Faster Using Lazy Global Domination

A clique in a graph is a set of vertices, each of which is adjacent to every other vertex in this set. A k-clique relaxes this requirement, requiring vertices to be within a distance k of each other, rather than directly adjacent. In theory, a maximum clique algorithm can easily be adapted to solve the maximum k-clique problem, although large sparse k-clique graphs reduce to large dense clique graphs, which can be computationally challenging. We adapt a state of the art maximum clique algorithm to show that this reduction is in fact useful in practice, and introduce a lazy global domination rule which sometimes vastly reduces the search space. We include experimental results for a range of real-world and benchmark graphs, and a detailed look at random graphs. We also use thread-parallel search to solve some harder instances.

Finding the shortest path for a Hypergraph

A hypergraph is given by [Formula: see text], where [Formula: see text] is a set of vertices and [Formula: see text] is a set of nonempty subsets of [Formula: see text], the member of [Formula: see text] is named hyperedge. So, a hypergraph is a nature generalization of a graph. A hypergraph has a complex structure, thus some researchers try to transform a hypergraph to a graph. In this paper, we define two graphs, Clique graph and Persian graph. These relations are one to one. We can find the shortest path between two vertices in a hypergraph [Formula: see text], by using the Dijkstra algorithm in graph theory on the graphs corresponding to [Formula: see text].

Strong Cliques in Claw-Free Graphs

For a graph G, L(G)^2 is the square of the line graph of G – that is, vertices of L(G)^2 are edges of G and two edges e,fin E(G) are adjacent in L(G)^2 if at least one vertex of e is adjacent to a vertex of f and ene f. The strong chromatic index of G, denoted by s'(G), is the chromatic number of L(G)^2. A strong clique in G is a clique in L(G)^2. Finding a bound for the maximum size of a strong clique in a graph with given maximum degree is a problem connected to a famous conjecture of Erdős and Nešetřil concerning strong chromatic index of graphs. In this note we prove that a size of a strong clique in a claw-free graph with maximum degree varDelta is at most varDelta ^2 + frac{1}{2}varDelta . This result improves the only known result 1.125varDelta ^2+varDelta , which is a bound for the strong chromatic index of claw-free graphs.

NI-Louvain: A novel algorithm to detect overlapping communities with influence analysis

Community detection is an important area of research in social media mining. Numerous algorithms have been developed to detect disjoint, overlapping, and dynamic communities in a network. However, most of the existing algorithms do not consider the influence of each node in a group, which may be different and can affect the result of community detection. In this paper, we have proposed a novel Louvain-based algorithm named “NI-Louvain,” which considers the influence of each node in a group that not only detects community but also most influential nodes in the community. The proposed algorithm works in three steps. First, the input graph is processed to reduce its density. This is done by calculating cliques from the graph. In second step, Louvain's multilevel algorithm is applied to this clique graph. The resultant community obtained has a better modularity value than the communities obtained from existing algorithms like edge betweenness, label propagation, leading eigen, Louvain, fast greedy, walktrap, and infomap. In the third step, the nodes of resultant communities are transformed back to the nodes of the original graph. This step is beneficial in detecting overlapping communities, fuzzy membership of each node, most influential node in each community formed, and outlier nodes.

Clique Is Hard on Average for Regular Resolution

We prove that for k ≪ 4√ n regular resolution requires length n Ω( k ) to establish that an Erdős–Rényi graph with appropriately chosen edge density does not contain a k -clique. This lower bound is optimal up to the multiplicative constant in the exponent and also implies unconditional n Ω( k ) lower bounds on running time for several state-of-the-art algorithms for finding maximum cliques in graphs.

Open Relation Extraction for Chinese Noun Phrases

Relation Extraction (RE) aims at harvesting relational facts from texts. A majority of existing research targets at knowledge acquisition from sentences, where subject-verb-object structures are usually treated as the signals of existence of relations. In contrast, relational facts expressed within noun phrases are highly implicit. Previous works mostly relies on human-compiled assertions and textual patterns in English to address noun phrase-based RE. For Chinese, the corresponding task is non-trivial because Chinese is a highly analytic language with flexible expressions. Additionally, noun phrases tend to be incomplete in grammatical structures, where clear mentions of predicates are often missing. In this article, we present an unsupervised Noun Phrase-based Open RE system for the Chinese language (NPORE), which employs a three-layer data-driven architecture. The system contains three components, i.e., Modifier-sensitive Phrase Segmenter, Candidate Relation Generator and Missing Relation Predicate Detector. It integrates with a graph clique mining algorithm to chunk Chinese noun phrases, considering how relations are expressed. We further propose a probabilistic method with knowledge priors and a hypergraph-based random walk process to detect missing relation predicates. Experiments over Chinese Wikipedia show NPORE outperforms state-of-the-art, capable of extracting 55.2 percent more relations than the most competitive baseline, with a comparable precision at 95.4 percent.

Line Graphs of Monogenic Semigroup Graphs

The concept of monogenic semigroup graphs Γ S M is firstly introduced by Das et al. (2013) based on zero divisor graphs. In this study, we mainly discuss the some graph properties over the line graph L Γ S M of Γ S M . In detail, we prove the existence of graph parameters, namely, radius, diameter, girth, maximum degree, minimum degree, chromatic number, clique number, and domination number over L Γ S M .

Ramsey-type numbers involving graphs and hypergraphs with large girth

AbstractErdős asked if, for every pair of positive integers g and k, there exists a graph H having girth (H) = k and the property that every r-colouring of the edges of H yields a monochromatic cycle Ck. The existence of such graphs H was confirmed by the third author and Ruciński.We consider the related numerical problem of estimating the order of the smallest graph H with this property for given integers r and k. We show that there exists a graph H on R10k2; k15k3 vertices (where R = R(Ck; r) is the r-colour Ramsey number for the cycle Ck) having girth (H) = k and the Ramsey property that every r-colouring of the edges of H yields a monochromatic Ck Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered.