Topic
Clique complex
About: Clique complex is a research topic. Over the lifetime, 76 publications have been published within this topic receiving 832 citations.
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TL;DR: Estimates suggest that almost all d-dimensional clique complexes have only one nonvanishing dimension of homology, and cannot rule out the possibility that they are homotopy equivalent to wedges of a spheres.
218 citations
TL;DR: This work formulate a two-phase approach for the Vietoris-Rips complex that separates geometry from topology, and gives three algorithms for the second phase, and implements all algorithms.
194 citations
TL;DR: The following extension of the Aharoni-Haxell matching Theorem is proved: Let be a family of hypergraphs such that for each either or , then there exists a matching such thatfor all .
Abstract: The width of a hypergraph is the minimal for which there exist such that for any , for some . The matching width of is the minimal such that for any matching there exist such that for any , for some . The following extension of the Aharoni-Haxell matching Theorem [3] is proved: Let be a family of hypergraphs such that for each either or , then there exists a matching such that for all . This is a consequence of a more general result on colored cliques in graphs. The proofs are topological and use the Nerve Theorem.
99 citations
01 Jan 2015
TL;DR: In this paper, the fundamental groups of clique complexes associated to random Erd˝ os-Renyi graphs were studied and thresholds for a number of properties of these complexes were established.
Abstract: We study fundamental groups of clique complexes associated to random Erd˝ os-Renyi graphs Γ We establish thresholds for a number of properties of fundamental groups of these complexes XΓ In particular, if p = n α , then we show that gdim(π1(XΓ)) = cd(π1(XΓ)) = 1 if α − 1 We prove that for − 11 <α< − 1 the fundamental group π1(XΓ) has 2-torsion but has no m-torsion for any given prime m 3 We also prove that aspherical subcomplexes of the random clique complex XΓ satisfy the Whitehead conjecture, that is, all their subcomplexes are also aspherical, aas
37 citations
TL;DR: The properties of spectra of combinatorial Laplacian operator of simplicial complexes are explored and its relationship with connectivity properties of the Q-vector and with connectivities of cliques in the simplicial clique complex is shown.
Abstract: Simplicial complexes represent useful and accurate models of complex networks and complex systems in general. We explore the properties of spectra of combinatorial Laplacian operator of simplicial complexes and show its relationship with connectivity properties of the Q-vector and with connectivities of cliques in the simplicial clique complex. We demonstrate the need for higher order analysis in complex networks and compare the results with ordinary graph spectra. Methods and results are obtained using social network of the Zachary karate club.
37 citations