Number Of Maximal Cliques Research Articles

Quantum circuits for discrete graphical models

Graphical models are useful tools for describing structured high-dimensional probability distributions. The development of efficient algorithms for generating samples thereof remains an active research topic. In this work, we provide a quantum algorithm that enables the generation of unbiased and independent samples from general discrete graphical models. To this end, we identify a coherent embedding of the graphical model based on a repeat-until-success sampling scheme which clearly identifies whether a drawn sample represents the underlying distribution. Furthermore, we show that the success probability for finding a valid sample can be lower bounded with a quantity that depends on the number of maximal cliques and the model parameter norm. Moreover, we rigorously proof that the quantum embedding conserves the key property of graphical models, i.e., factorization over the cliques of the underlying conditional independence structure. The quantum sampling algorithm also allows for maximum likelihood learning as well as maximum a posteriori state approximation for the graphical model. Finally, the proposed quantum method shows potential to realize interesting problems on near-term quantum processors. In fact, illustrative experiments demonstrate that our method can carry out sampling and parameter learning not only with idealized simulations of quantum computers but also on actual quantum hardware solely supported by simple readout error mitigation.

Accelerating Maximal Clique Enumeration via Graph Reduction

As a fundamental task in graph data management, maximal clique enumeration (MCE) has attracted extensive attention from both academic and industrial communities due to its wide range of applications. However, MCE is very challenging as the number of maximal cliques may grow exponentially with the number of vertices. The state-of-the-art methods adopt a recursive paradigm to enumerate maximal cliques exhaustively, suffering from a large amount of redundant computation. In this paper, we propose a novel reduction-based framework for MCE, namely RMCE, that aims to reduce the search space and minimize unnecessary computations. The proposed framework RMCE incorporates three kinds of powerful reduction techniques including global reduction, dynamic reduction, and maximality check reduction. Global and dynamic reduction techniques effectively reduce the size of the input graph and dynamically construct subgraphs during the recursive sub-tasks, respectively. The maximality check reduction minimizes the computation for ensuring maximality by utilizing neighborhood dominance between visited vertices. Extensive experiments on 18 real graphs demonstrate that the existing approaches achieve remarkable speedups powered by the proposed techniques.

The Maximum Number of Cliques in Graphs with Bounded Odd Circumference

AbstractIn this work, we give the sharp upper bound for the number of cliques in graphs with bounded odd circumferences. This generalized Turán-type result is an extension of the celebrated Erdős and Gallai theorem and a strengthening of Luo’s recent result. The same bound for graphs with bounded even circumferences is a trivial application of the theorem of Li and Ning.

Families of Disjoint Sets Colouring Technique and Concept of Common and Non Common Vertex

Families of disjoint sets colouring technique is trial to generalize all type of colouring, such as edge colouring, vertex colouring, face colouring, sets colouring (partition of sets into families of disjoint sets). In this paper we introduce the concept of common vertex between two cliques and more than two cliques. We introduce some results related to concept of common and non common vertex, and introduce some results explain the number of minimum colour classes not changed after addition or after removal of non common vertex, if maximum clique number is constant. Also in this paper we introduce concept of quasi non common vertex between two cliques and more than two cliques and we introduce some results related to this concept.

Finding biclique partitions of co-chordal graphs

The biclique partition number (bp) of a graph G is referred to as the least number of complete bipartite (biclique) subgraphs that are required to cover the edges of the graph exactly once. In this paper, we show that the biclique partition number of a co-chordal (complementary graph of chordal) graph G=(V,E) is less than the number of maximal cliques (mc) of its complementary graph: A chordal graph Gc=(V,Ec). We first provide a general framework of the “divide and conquer” heuristic of finding minimum biclique partitions of co-chordal graphs based on clique trees. Furthermore, a heuristic of complexity O(|V|(|V|+|Ec|)) is proposed by applying a lexicographic breadth-first search to find structures called moplexes. Either heuristic gives us a biclique partition of G with a size of mc(Gc)−1. In addition, we prove that both of our heuristics can solve the minimum biclique partition problem on G exactly if its complement Gc is chordal and clique vertex irreducible. We also show that mc(Gc)−2≤bp(G)≤mc(Gc)−1 if G is a split graph.

The maximum number of cliques in graphs with prescribed order, circumference and minimum degree

Erdős determined the maximum size of a nonhamiltonian graph of order n and minimum degree at least k in 1962. Recently, Ning and Peng generalized Erdős’s work and gave the maximum number of s-cliques hs(n,c,k) of graphs with prescribed order n, circumference c and minimum degree at least k. But for some quadruples n,c,k,s the maximum number of s-cliques is not attained by a graph of minimum degree k. For example, h2(15,14,3)=77 is attained by a unique graph of minimum degree 7, not 3. In this paper we obtain more precise information by determining the maximum number of s-cliques of a graph with prescribed order, circumference and minimum degree. As a corollary of our main result, we determine the maximum size of a k-connected graph with given order and circumference. Consequently we solve the corresponding problem for longest paths.

Counting cliques in 1-planar graphs

The problem of maximising the number of cliques among n-vertex graphs from various graph classes has received considerable attention. We investigate this problem for the class of 1-planar graphs where we determine precisely the maximum total number of cliques as well as the maximum number of cliques of any fixed size. We also precisely characterise the extremal graphs for these problems.

On the overall and delay complexity of the CLIQUES and Bron-Kerbosch algorithms

We revisit the maximal clique enumeration algorithm cliques by Tomita et al. that appeared in Theoretical Computer Science in 2006. It is known to work in O(3n/3)-time in the worst-case for an n-vertex graph. This is worst-case optimal with respect to the input size, but there is little knowledge about its performance with respect to the output. In this paper, we extend the time-complexity analysis with respect to the maximum size and the number of maximal cliques, and to its delay, solving issues that were left as open problems since the original paper. In particular, we prove that cliques has Ω(3n/6) delay and that, even if we allow to change the pivoting strategy, a variant having polynomial delay cannot be designed unless P=NP. These same results apply to the related Bron-Kerbosch algorithm. On the positive side, we show that the complexity of cliques and Bron-Kerbosch is amortized polynomial on graphs with logarithmic clique number. As these algorithms are widely used and regarded as fast “in practice”, we are interested in observing their practical behavior: we run an evaluation of cliques and three Bron-Kerbosch variants on over 130 real-world and synthetic graphs, observing how the clique number almost always satisfies our logarithmic constraint, and that their performance seems far from its theoretical worst-case behavior in terms of both total time and delay.1

Many cliques with few edges and bounded maximum degree

Generalized Turán problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem, maximizing the number of cliques of a fixed order in a graph with fixed number of vertices and bounded maximum degree, was recently completely resolved by Chase. Kirsch and Radcliffe raised a natural variant of this problem where the number of edges is fixed instead of the number of vertices. In this paper, we determine the maximum number of cliques of a fixed order in a graph with fixed number of edges and bounded maximum degree, resolving a conjecture by Kirsch and Radcliffe. We also give a complete characterization of the extremal graphs.

Predicting Brain Regions Related to Alzheimer's Disease Based on Global Feature.

Alzheimer's disease (AD) is a neurodegenerative disease that commonly affects the elderly; early diagnosis and timely treatment are very important to delay the course of the disease. In the past, most brain regions related to AD were identified based on imaging methods, and only some atrophic brain regions could be identified. In this work, the authors used mathematical models to identify the potential brain regions related to AD. In this study, 20 patients with AD and 13 healthy controls (non-AD) were recruited by the neurology outpatient department or the neurology ward of Peking University First Hospital from September 2017 to March 2019. First, diffusion tensor imaging (DTI) was used to construct the brain structural network. Next, the authors set a new local feature index 2hop-connectivity to measure the correlation between different regions. Compared with the traditional graph theory index, 2hop-connectivity exploits the higher-order information of the graph structure. And for this purpose, the authors proposed a novel algorithm called 2hopRWR to measure 2hop-connectivity. Then, a new index global feature score (GFS) based on a global feature was proposed by combing five local features, namely degree centrality, betweenness centrality, closeness centrality, the number of maximal cliques, and 2hop-connectivity, to judge which brain regions are related to AD. As a result, the top ten brain regions identified using the GFS scoring difference between the AD and the non-AD groups were associated to AD by literature verification. The results of the literature validation comparing GFS with the local features showed that GFS was superior to individual local features. Finally, the results of the canonical correlation analysis showed that the GFS was significantly correlated with the scores of the Mini-Mental State Examination (MMSE) scale and the Montreal Cognitive Assessment (MoCA) scale. Therefore, the authors believe the GFS can also be used as a new biomarker to assist in diagnosis and objective monitoring of disease progression. Besides, the method proposed in this paper can be used as a differential network analysis method for network analysis in other domains.

Some classes of Trivially Perfect Graphs

A graph G is supposed to be trivially perfect if, in each induced subgraph H of G, the number of maximal cliques in H equivalents to the size of a maximum independent set in H. Trivially perfect graphs is subclasses of notable perfect graphs and its characterization have numerous continuous applications and it is adequate to research its subclasses. Along with this idea, in this paper, it is discussed trivially perfect graphs on the windmill graph and demonstrated a few outcomes on trivially perfect graphs.

The Maximum Number of Cliques in Hypergraphs without Large Matchings

Let $[n]$ denote the set $\{1, 2, \ldots, n\}$ and $\mathcal{F}^{(r)}_{n,k,a}$ be an $r$-uniform hypergraph on the vertex set $[n]$ with edge set consisting of all the $r$-element subsets of $[n]$ that contains at least $a$ vertices in $[ak+a-1]$. For $n\geq 2rk$, Frankl proved that $\mathcal{F}^{(r)}_{n,k,1}$ maximizes the number of edges in $r$-uniform hypergraphs on $n$ vertices with the matching number at most $k$. Huang, Loh and Sudakov considered a multicolored version of the Erd\H{o}s matching conjecture, and provided a sufficient condition on the number of edges for a multicolored hypergraph to contain a rainbow matching of size $k$. In this paper, we show that $\mathcal{F}^{(r)}_{n,k,a}$ maximizes the number of $s$-cliques in $r$-uniform hypergraphs on $n$ vertices with the matching number at most $k$ for sufficiently large $n$, where $a=\lfloor \frac{s-r}{k} \rfloor+1$. We also obtain a condition on the number of $s$-clques for a multicolored $r$-uniform hypergraph to contain a rainbow matching of size $k$, which reduces to the condition of Huang, Loh and Sudakov when $s=r$.

Improved community structure discovery algorithm based on combined clique percolation method and K-means algorithm

Research on the community structure of networks is beneficial for understanding the structure of networks, analyzing their characteristics and discovering the rules hidden in these networks. To address issues from previous community mining algorithms, such as the low rate of convergence and high time complexity, this study proposes an improved community structure discovery algorithm named CPMK-Means algorithm. The main idea of this algorithm can be summarised as follows. The clique percolation method (CPM) algorithm generates the maximum number of cliques by combining depth-first search with breadth-first search so that the number of cluster centres is determined. Then, the k centres are selected based on the principle of the maximum degree of centres and minimum similarity between different centres. Afterwards, nodes in the network are assigned to the communities formed by the k centres, and the iterations are performed repeatedly until the centres become stable. Finally, the overlapping communities are merged. Experiments are carried out on standard data sets Football and Collins to evaluate the performance of the CPMK-Means algorithm. Results indicate that the CPMK-Means algorithm can achieve better community mining and higher execution efficiency compared with other algorithms. Furthermore, it is superior to other algorithms in terms of precision, recall, accuracy, F-measure and separation.

A variation of a theorem by Pósa

A graph G is ℓ-hamiltonian if each linear forest F with ℓ edges contained in G can be extended to a hamiltonian cycle of G. We give a sharp upper bound for the maximum number of cliques of a fixed size in a non-ℓ-hamiltonian graph. Furthermore, we prove stability: if a non-ℓ-hamiltonian graph contains almost the maximum number of cliques, then it is a subgraph of one of two extremal graphs.

The number of maximal cliques and spectral radius of graphs with certain forbidden subgraphs

Turán theorem states that the Turán graph [Formula: see text] is the unique graph which has the maximum edge number among the [Formula: see text]-free graphs of order [Formula: see text]. In this paper, we prove that [Formula: see text] has both the maximum number of maximal cliques and the maximum spectral radius among all graphs of order [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] stands for the maximum number of disjoint [Formula: see text]cliques of [Formula: see text].

On the Logistic Behaviour of the Topological Ultrametricity of Data

Recently, it has been observed that topological ultrametricity of data can be expressed as an integral over a function which describes local ultrametricity. It was then observed empirically that this function begins as a sharply decreasing function, in order to increase again back to one. After providing a method for estimating the falling part of the local ultrametricity of data, empirical evidence is given for its logistic behaviour in relation to the number of connected components of the Vietoris-Rips graphs involved. The result is a functional dependence between that number and the number of maximal cliques. Further, it turns out that the logistic parameters depend linearly on the datasize. These observations are interpreted in terms of the Erdős-Renyi model for random graphs. Thus the findings allow to define a percolationbased index for almost ultrametricity which can be estimated in O(N2 logN) time which is more efficient than most ultrametricity indices.

On the nullity number of graphs

The paper discusses bounds on the nullity number of graphs. It is proved in [B. Cheng and B. Liu, On the nullity of graphs. Electron. J. Linear Algebra 16 (2007) 60--67] that $\eta \le n - D$, where $\eta$, n and D denote the nullity number, the order and the diameter of a connected graph, respectively. We first give a necessary condition on the extremal graphs corresponding to that bound, and then we strengthen the bound itself using the maximum clique number. In addition, we prove bounds on the nullity using the number of pendant neighbors in a graph. One of those bounds is an improvement of a known bound involving the domination number.

Parameterized algorithms for recognizing monopolar and 2-subcolorable graphs

A graph G is a (ΠA,ΠB)-graph if V(G) can be bipartitioned into A and B such that G[A] satisfies property ΠA and G[B] satisfies property ΠB. The (ΠA,ΠB)-Recognition problem is to recognize whether a given graph is a (ΠA,ΠB)-graph. There are many (ΠA,ΠB)-Recognition problems, including the recognition problems for bipartite, split, and unipolar graphs. We present efficient algorithms for many cases of (ΠA,ΠB)-Recognition based on a technique which we dub inductive recognition. In particular, we give fixed-parameter algorithms for two NP-hard (ΠA,ΠB)-Recognition problems, Monopolar Recognition and 2-Subcoloring, parameterized by the number of maximal cliques in G[A]. We complement our algorithmic results with several hardness results for (ΠA,ΠB)-Recognition.

The maximum number of cliques in graphs without long cycles

The Erdős–Gallai Theorem states that for k≥3 every graph on n vertices with more than 12(k−1)(n−1) edges contains a cycle of length at least k. Kopylov proved a strengthening of this result for 2-connected graphs with extremal examples Hn,k,t and Hn,k,2. In this note, we generalize the result of Kopylov to bound the number of s-cliques in a graph with circumference less than k. Furthermore, we show that the same extremal examples that maximize the number of edges also maximize the number of cliques of any fixed size. Finally, we obtain the extremal number of s-cliques in a graph with no path on k-vertices.

On the number of cliques in graphs with a forbidden minor

Reed and Wood and independently Norine, Seymour, Thomas, and Wollan proved that for each positive integer t there is a constant c(t) such that every graph on n vertices with no Kt-minor has at most c(t)n cliques. Wood asked in 2007 if we can take c(t)=ct for some absolute constant c. This question was recently answered affirmatively by Lee and Oum. In this paper, we determine the exponential constant. We prove that every graph on n vertices with no Kt-minor has at most 32t/3+o(t)n cliques. This bound is tight for n≥4t/3. More generally, let H be a connected graph on t vertices, and x denote the size (i.e., the number edges) of the largest matching in the complement of H. We prove that every graph on n vertices with no H-minor has at most max⁡(32t/3−x/3+o(t)n,2t+o(t)n) cliques, and this bound is tight for n≥max⁡(4t/3−2x/3,t) by a simple construction. Even more generally, we determine explicitly the exponential constant for the maximum number of cliques an n-vertex graph can have in a minor-closed family of graphs which is closed under disjoint union.