Maximal Complete Subgraphs Research Articles

Characterising clique convergence for locally cyclic graphs of minimum degree δ ≥ 6

The clique graph kG of a graph G has as its vertices the cliques (maximal complete subgraphs) of G, two of which are adjacent in kG if they have non-empty intersection in G. We say that G is clique convergent if knG≅kmG for some n≠m, and that G is clique divergent otherwise. We completely characterise the clique convergent graphs in the class of (not necessarily finite) locally cyclic graphs of minimum degree δ≥6, showing that for such graphs clique divergence is a global phenomenon, dependent on the existence of large substructures. More precisely, we establish that such a graph is clique divergent if and only if its universal triangular cover contains arbitrarily large members from the family of so-called “triangular-shaped graphs”.

Nourishing Number of Some Associated Graphs

Let N0 = N∪{0} and P(N0) be the power set. An injection f : V (G) → P(N0) is an integer additive set-indexer (IASI) of a graph G if the induced map f+ : E(G) → P(N0) given by f+(uv) = f(u) + f(v) is also an injection, where f(u) + f(v) is the sumset of f(u) and f(v). Moreover, if |f+(uv)| = |f(u)| |f(v)|, for all uv in E(G), then f is a strong IASI of G. The nourishing number of a graph G is the minimum order of the maximal complete subgraph of G such that G admits a strong IASI. In this paper we investigate the admissibility of strong IASI for some associated graphs and calculate their nourishing number. In addition, we obtain the nourishing number of powers of the associated graphs.

On the clique behavior and Hellyness of the complements of regular graphs

A collection of sets is intersecting, if any pair of sets in the collection has nonempty intersection. A collection of sets has the Helly property if any intersecting subcollection has nonempty intersection. A graph G is Helly if the collection of maximal complete subgraphs of G has the Helly property. We prove that if G is a k-regular graph with n vertices such that n>3k+2k2−k, then the complement G‾ is not Helly. We also consider the problem of whether the properties of Hellyness and convergence under the clique graph operator are equivalent for the complement of k-regular graphs, for small values of k.

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.

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.

Edge erasures and chordal graphs

We prove several results about chordal graphs and weighted chordal graphs by focusing on exposed edges. These are edges that are properly contained in a single maximal complete subgraph. This leads to a characterization of chordal graphs via deletions of a sequence of exposed edges from a complete graph. Most interesting is that in this context the connected components of the edge-induced subgraph of exposed edges are 2 -edge connected. We use this latter fact in the weighted case to give a modified version of Kruskal's second algorithm for finding a minimum spanning tree in a weighted chordal graph. This modified algorithm benefits from being local in an important sense.

Biclique Graphs of K3-free Graphs and Bipartite Graphs

A biclique of a graph is a maximal complete bipartite subgraph. The biclique graph of a graph G, KB(G), defined as the intersection graph of the bicliques of G, was introduced and characterized in 2010 by Groshaus and Szwarcfiter. However, this characterization does not lead to polynomial time recognition algorithms, and the time complexity of its recognition problem remains open. There are some works on this problem when restricted to some classes. In this work we give a characterization of the biclique graph of a K3-free graph G. We prove that KB(G) is the square graph of a particular graph which we call Mutually Included Biclique Graph of G, KBm(G). Although it does not lead to a polynomial time recognition algorithm, it gives a new tool to prove properties of biclique graphs (restricted to K3-free graphs) using known properties of square graphs. For instance we generalize a property about induced P3's in biclique graphs to a property about stars and proved a conjecture posted by Groshaus and Montero, when restricted to K3-free graphs. Also we give another characterization of the class of biclique graphs of bipartite graphs. We prove that KB(bipartite) = (IIC-comparability)2, where IIC-comparability is a subclass of comparability graphs that we call Interval Intersection Closed Comparability.

A graph density-based strategy for features fusion from different peak extract software to achieve more metabolites in metabolic profiling from high-resolution mass spectrometry

In metabolomics study, it is not easy to extract the metabolites from data of ultra high-performance liquid chromatography-high-resolution mass spectrometry, especially for those with low abundance. Different software for peak recognition and matching use different algorithms, leading to different extract results. Therefore, integration of results from different software can obtain richer metabolome information, but the redundant features should be removed. In this study, an integrated strategy of fusing features and removing redundancy based on graph density (FRRGD) was proposed. A graph is used to cover the ion features generated by two open access software (XCMS, MZmine 2) and a software (SIEVE) from an instrument vendor, and redundant features were removed by searching the maximal complete sub-graphs. A standard mixture containing 41 metabolites and a spontaneous urine were utilized to develop the method and demonstrate its usefulness. For the standard mixture, 19, 19 and 27 metabolites were extracted by XCMS, MZmine 2 and SIEVE, respectively. After fusion by FRRGD, 37 metabolites were obtained. For the diluted spontaneous urine sample, 1103, 1500 and 387 metabolites were extracted by XCMS, MZmine 2 and SIEVE, respectively, FRRGD produced 1619 metabolites which were much more than individual software, significantly increasing metabolome coverage. The proposed FRRGD shows a great prospect as a new data processing strategy for metabolomics study.

Detecting overlapping protein complexes in weighted PPI network based on overlay network chain in quotient space

BackgroundProtein complexes are the cornerstones of many biological processes and gather them to form various types of molecular machinery that perform a vast array of biological functions. In fact, a protein may belong to multiple protein complexes. Most existing protein complex detection algorithms cannot reflect overlapping protein complexes. To solve this problem, a novel overlapping protein complexes identification algorithm is proposed.ResultsIn this paper, a new clustering algorithm based on overlay network chain in quotient space, marked as ONCQS, was proposed to detect overlapping protein complexes in weighted PPI networks. In the quotient space, a multilevel overlay network is constructed by using the maximal complete subgraph to mine overlapping protein complexes. The GO annotation data is used to weight the PPI network. According to the compatibility relation, the overlay network chain in quotient space was calculated. The protein complexes are contained in the last level of the overlay network. The experiments were carried out on four PPI databases, and compared ONCQS with five other state-of-the-art methods in the identification of protein complexes.ConclusionsWe have applied ONCQS to four PPI databases DIP, Gavin, Krogan and MIPS, the results show that it is superior to other five existing algorithms MCODE, MCL, CORE, ClusterONE and COACH in detecting overlapping protein complexes.

Further developments on Brouwer's conjecture for the sum of Laplacian eigenvalues of graphs

Let G be a simple graph with order n and size m and having Laplacian eigenvalues μ1,μ2,…,μn−1,μn=0 and let Sk(G)=∑i=1kμi be the sum of k largest Laplacian eigenvalues of G. Brouwer conjectured that Sk(G)≤m+(k+12), for all k=1,2,…,n. We obtain upper bounds for Sk(G), in terms of the clique number ω, the order n and integers p≥0,r≥1,s1≥s2≥2 associated to the structure of the graph G. We discuss Brouwer's conjecture for two large families of graphs; the first family of graphs is obtained from a clique of size ω by identifying each of its vertices to a vertex of an arbitrary c-cyclic graph, and the second family is composed of the graphs in which the removal of the edges of the maximal complete bipartite subgraph gives a graph each of whose non-trivial components is a c-cyclic graph. We show among these two large families of graphs, the Brouwer's conjecture holds for various subfamilies of graphs depending upon the value of c, the order of the c-cyclic graphs, the clique number of the graph, the order of the maximal complete bipartite subgraph and the number of the c-cyclic components of the graph.

Listing all maximal cliques in large graphs on vertex-centric model

Maximal Clique Enumeration (MCE), which consists to enumerate all maximal complete subgraphs in a given graph, is a fundamental problem in graph theory, and it is used in several applications. It is one of Karp’s 21 NP-complete problems. In the literature, this problem has been widely studied. One of the most notable, efficient, successful and extensively used solutions is the Bron–Kerbosch (BK) algorithm. The latter is a sequential algorithm which is able to enumerate all maximal cliques in an undirected graph without duplication. Furthermore, it is used to solve large problems such as maximum clique problem, community detection and graph clustering. However, for large graphs, sequential algorithms are slow and do not scale well. Thus, processing efficiently this kind of graphs needs to develop distributed algorithms under parallel and distributed platforms or large graph mining frameworks. In this setting, we propose new efficient distributed algorithms for maximal clique enumerating based on the vertex-centric model. These algorithms use the BK algorithm principle to deal with the MCE problem on large cluster. The proposed algorithms are implemented in Giraph and evaluated by using real-world graphs and computer-generated benchmark networks. Our experiments on a Hadoop cluster show that the proposed algorithms can effectively process a variety of large real-world and computer-generated graphs and scale well with increasing the dataset size and the number of nodes in the cluster. Furthermore, the proposed algorithms are provably work-efficient compared with other algorithms including BK algorithm.

Fold change based approach for identification of significant network markers in breast, lung and prostate cancer.

Cancer belongs to a class of highly aggressive diseases and a leading cause of death in the world. With more than 100 types of cancers, breast, lung and prostate cancer remain to be the most common types. To identify essential network markers (NMs) and therapeutic targets in these cancers, the authors present a novel approach which uses gene expression data from microarray and RNA‐seq platforms and utilises the results from this data to evaluate protein–protein interaction (PPI) network. Differentially expressed genes (DEGs) are extracted from microarray data using three different statistical methods in R, to produce a consistent set of genes. Also, DEGs are extracted from RNA‐seq data for the same three cancer types. DEG sets found to be common in both platforms are obtained at three fold change (FC) cut‐off levels to accurately identify the level of change in expression of these genes in all three cancers. A cancer network is built using PPI data characterising gene sets at log‐FC (LFC)>1, LFC>1.5 and LFC>2, and interconnection between principal hub nodes of these networks is observed. Resulting network of hubs at three FC levels highlights prime NMs with high confidence in multiple cancers as validated by Gene Ontology functional enrichment and maximal complete subgraphs from CFinder.

Complete subgraphs of the coprime hypergraph of integers II: structural properties

We continue to investigate structural properties of complete subgraphs of the coprime hypergraph of integers on n vertices , which has vertex set and hyperedge set . It turns out that any maximal complete subgraph G of has only vertices with at most three prime divisors for sufficiently large n. Further, we show that a vertex a of G with three prime divisors is squarefree and satisfies . We also give upper bounds for the number of prime divisors of vertices of G in case of arbitrary n.

Binomial partial Steiner triple systems containing complete graphs

We propose a new approach to studies on partial Steiner triple systems consisting in determining complete graphs contained in them. We establish the structure which complete graphs yield in a minimal PSTS that contains them. As a by-product we introduce the notion of a binomial PSTS as a configuration with parameters of a minimal PSTS with a complete subgraph. A representation of binomial PSTS with at least a given number of its maximal complete subgraphs is given in terms of systems of perspectives. Finally, we prove that for each admissible integer there is a binomial PSTS with this number of maximal complete subgraphs.

On self-clique graphs with triangular cliques

A graph is an {r,s}-graph if the set of degrees of their vertices is {r,s}. A clique of a graph is a maximal complete subgraph. The clique graphK(G) of a graph G is the intersection graph of all its cliques. A graph G is self-clique if G is isomorphic to K(G). We show the existence of self-clique {5,6}-graphs whose cliques are all triangles, thus solving a problem posed by Chia and Ong (2012).

A Study on the Nourishing Number of Graphs and Graph Powers

Let \(\mathbb{N}_{0}\) be the set of all non-negative integers and \(\mathcal{P}(\mathbb{N}_{0})\) be its power set. Then, an integer additive set-indexer (IASI) of a given graph \(G\) is defined as an injective function \(f:V(G)\to \mathcal{P}(\mathbb{N}_{0})\) such that the induced edge-function \(f^+:E(G) \to\mathcal{P}(\mathbb{N}_{0})\) defined by \(f^+ (uv) = f(u)+ f(v)\) is also injective, where \(f(u)+f(v)\) is the sumset of \(f(u)\) and \(f(v)\). An IASI \(f\) of \(G\) is said to be a strong IASI of \(G\) if \(|f^+(uv)|=|f(u)|\,|f(v)|\) for all \(uv\in E(G)\). The nourishing number of a graph \(G\) is the minimum order of the maximal complete subgraph of \(G\) so that \(G\) admits a strong IASI. In this paper, we study the characteristics of certain graph classes and graph powers that admit strong integer additive set-indexers and determine their corresponding nourishing numbers.

Cohen-Macaulay bipartite graphs in arbitrary codimension

Let G G be an unmixed bipartite graph of dimension d − 1 d-1 . Assume that K n , n K_{n,n} , with n ≥ 2 n\ge 2 , is a maximal complete bipartite subgraph of G G of minimum dimension. Then G G is Cohen-Macaulay in codimension t t if and only if t ≥ d − n + 1 t\ge d-n+1 . This is derived from a characterization of Cohen-Macaulay bipartite graphs by Herzog and Hibi and generalizes a recent result of Cook and Nagel on unmixed Buchsbaum graphs. Furthermore, we show that any unmixed bipartite graph G G which is Cohen-Macaulay in codimension t t , is obtained from a Cohen-Macaulay graph by replacing certain edges of G G with complete bipartite graphs. Thus, in light of combinatorial characterization of Cohen-Macaulay bipartite graphs, our result may be considered purely combinatorial.

Biclique-colouring verification complexity and biclique-colouring power graphs

Biclique-colouring is a colouring of the vertices of a graph in such a way that no maximal complete bipartite subgraph with at least one edge is monochromatic. We show that it is coNP-complete to check whether a given function that associates a colour to each vertex is a biclique-colouring, a result that justifies the search for structured classes where the biclique-colouring problem could be efficiently solved. We consider biclique-colouring restricted to powers of paths and powers of cycles. We determine the biclique-chromatic number of powers of paths and powers of cycles. The biclique-chromatic number of a power of a path Pnk is max(2k+2−n,2) if n≥k+1 and exactly n otherwise. The biclique-chromatic number of a power of a cycle Cnk is at most 3 if n≥2k+2 and exactly n otherwise; we additionally determine the powers of cycles that are 2-biclique-colourable. All proofs are algorithmic and provide polynomial-time biclique-colouring algorithms for graphs in the investigated classes.

Predicting disease-related proteins based on clique backbone in protein-protein interaction network.

Network biology integrates different kinds of data, including physical or functional networks and disease gene sets, to interpret human disease. A clique (maximal complete subgraph) in a protein-protein interaction network is a topological module and possesses inherently biological significance. A disease-related clique possibly associates with complex diseases. Fully identifying disease components in a clique is conductive to uncovering disease mechanisms. This paper proposes an approach of predicting disease proteins based on cliques in a protein-protein interaction network. To tolerate false positive and negative interactions in protein networks, extending cliques and scoring predicted disease proteins with gene ontology terms are introduced to the clique-based method. Precisions of predicted disease proteins are verified by disease phenotypes and steadily keep to more than 95%. The predicted disease proteins associated with cliques can partly complement mapping between genotype and phenotype, and provide clues for understanding the pathogenesis of serious diseases.

Large collections of curves pairwise intersecting exactly once

Let \(\Omega =(\omega _{j})_{j\in I}\) be a maximum size collection of pairwise non-isotopic simple closed curves on the closed, orientable, genus \(g\) surface \(S_{g}\), such that \(\omega _{i}\) and \(\omega _{j}\) intersect exactly once for \(i\ne j\). We show that for \(g\ge 3\), there exists atleast two such collections up to the action of the mapping class group, answering a question posed by Malestein, Rivin and Theran. As a consequence, we show that the automorphism group of the systole graph for \(S_{g}, g\ge 3\) (whose vertices are isotopy classes of simple closed curves, and whose edges correspond to pairs of curve intersecting once) does not act transitively on maximal complete subgraphs.