Topology of clique complexes of line graphs

Goyal, Shuchita; Shukla, Samir; Singh, Anurag K.

doi:10.26493/2590-9770.1434.bf4

Cited by 2 publications

(2 citation statements)

References 17 publications

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“…The following result is proved in [9]. This is an important method to investigate the homotopy type of a complex by splitting it into two or more subcomplexes.…”

Section: Notations and Preliminary Resultsmentioning

confidence: 97%

On Vietoris-Rips complexes of Finite Metric Spaces with Scale $2$

Zhao¹,

Nukala²

2023

Preprint

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We examine the homotopy types of Vietoris-Rips complexes on certain finite metric spaces at scale 2. We consider the collections of subsets of [m] = {1, 2, . . . , m} equipped with symmetric difference metric d, specifically,Here F m n is the collection of size n subsets of [m] and F m A is the collection of subsets A where is a total order on the collections of subsets of [m] and A ⊆ [m] (see the definition of in Section 1). We prove that the Vietoris-Rips complexes VR(F m n , 2) and F m n ∪ F m n+1 are either contractible or homotopy equivalent to a wedge sum of S 2 's; also, the complexes VR(F m n ∪ F m n+2 , 2) and VR(F m A , 2) are either contractible or homotopy equivalent to a wedge sum of S 3 's. We provide inductive formula for these homotopy types extending the result of Barmak in [4] about the independence complexes of Kneser graphs KG 2,k and the result of Adamamszek and Adams in [2] about Vietoris-Rips complexes of hypercube graphs with scale 2.

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“…The following result is proved in [9]. This is an important method to investigate the homotopy type of a complex by splitting it into two or more subcomplexes.…”

Section: Notations and Preliminary Resultsmentioning

confidence: 97%

On Vietoris-Rips complexes of Finite Metric Spaces with Scale $2$

Zhao¹,

Nukala²

2023

Preprint

View full text Add to dashboard Cite

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“…For example, he used star clusters to show that the independence complex of any triangle-free graph has the homotopy type of a suspension and that the independence complex of a forest is either contractible or homotopy equivalent to a sphere. K. Iriye used star clusters to construct a matching tree for the independence complex of square grids with cyclic identification [8], and S. Goyal et al used star clusters to compute the homotopy type of the independence complexes of generalised Mycielskian of complete graphs [6]. Another important tool in combinatorial topology is the Cluster lemma.…”

Section: Introductionmentioning

confidence: 99%

Star clusters in the Matching, Morse, and Generalized complex of discrete Morse functions

Donovan

Scoville

2023

Preprint

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In this paper, we determine the homotopy type of the complex of discrete Morse functions and matching complex of multiple families of complexes by utilizing star cluster collapses and the Cluster Lemma. We compute the homotopy type of the complex of discrete Morse functions of an extended notion of a star graph, as well as the homotopy type of the matching complex of a Dutch windmill graph. Additionally, we provide alternate computations of the homotopy type of the complex of discrete Morse functions of paths, the homotopy type of the matching complex of paths, and the homotopy type of the matching complex of cycles. We then use this same method of computing homotopy types to investigate the relationship between the homotopy type of the matching complex and the generalized Morse complex. 2020 Mathematics Subject Classification. (Primary) 57Q70; (Secondary) 55P10, 55U10, 05C70.

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Topology of clique complexes of line graphs

Cited by 2 publications

References 17 publications

On Vietoris-Rips complexes of Finite Metric Spaces with Scale $2$

On Vietoris-Rips complexes of Finite Metric Spaces with Scale $2$

Star clusters in the Matching, Morse, and Generalized complex of discrete Morse functions

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