The authors introduce a notion of a weak graph map homotopy(they call it M-homotopy),discuss its properties and applications.They prove that the weak graph map homotopy equivalence between graphs coincides with the gr...The authors introduce a notion of a weak graph map homotopy(they call it M-homotopy),discuss its properties and applications.They prove that the weak graph map homotopy equivalence between graphs coincides with the graph homotopy equivalence defined by Yau et al in 2001.The difference between them is that the weak graph map homotopy transformation is defined in terms of maps,while the graph homotopy transformation is defined by means of combinatorial operations.They discuss its advantages over the graph homotopy transformation.As its applications,they investigate the mapping class group of a graph and the 1-order M P-homotopy group of a pointed simple graph.Moreover,they show that the 1-order M P-homotopy group of a pointed simple graph is invariant up to the weak graph map homotopy equivalence.展开更多
In this paper, the authors study further properties and applications of weighted homology and persistent homology. The Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology are intr...In this paper, the authors study further properties and applications of weighted homology and persistent homology. The Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology are introduced. For applications, the authors show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. They also prove a theorem to calculate the mod p^(2) weighted persistent homology provided with some information on the mod p weighted persistent homology.展开更多
基金supported by the National Natural Science Foundation of China(No.11771116)。
文摘The authors introduce a notion of a weak graph map homotopy(they call it M-homotopy),discuss its properties and applications.They prove that the weak graph map homotopy equivalence between graphs coincides with the graph homotopy equivalence defined by Yau et al in 2001.The difference between them is that the weak graph map homotopy transformation is defined in terms of maps,while the graph homotopy transformation is defined by means of combinatorial operations.They discuss its advantages over the graph homotopy transformation.As its applications,they investigate the mapping class group of a graph and the 1-order M P-homotopy group of a pointed simple graph.Moreover,they show that the 1-order M P-homotopy group of a pointed simple graph is invariant up to the weak graph map homotopy equivalence.
基金This work was supported by the Singapore Ministry of Education Research Grant(AcRF Tier 1 WBS No.R-146-000-222-112)the Postdoctoral International Exchange Program of China 2019 Project from the Office of China Postdoctoral Council+4 种基金China Postdoctoral Science Foundationthe President’s Graduate Fellowship of National University of Singaporethe Natural Science Foundation of China(Nos.11971144,12001310)High-Level Scientific Research Foundation of Hebei ProvinceChina Postdoctoral Science Foundation(No.2019-2021)。
文摘In this paper, the authors study further properties and applications of weighted homology and persistent homology. The Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology are introduced. For applications, the authors show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. They also prove a theorem to calculate the mod p^(2) weighted persistent homology provided with some information on the mod p weighted persistent homology.
