Homotopy theory of graphs
نویسندگان
چکیده
Recently a new homotopy theory for graphs and simplicial complexes was defined (cf. [3, 4]). The motivation for the definition came initially from a desire to find invariants for dynamic processes that could be encoded via (combinatorial) simplicial complexes. The invariants were supposed to be topological in nature, but should at the same time be sensitive to the combinatorics encoded in the complex, in particular to the level of connectivity of simplices (see [7]). Namely, let be a simplicial complex of dimension d, let 0 ≤ q ≤ d be an integer, and let σ0 ∈ be a simplex of dimension greater than or equal to q . One obtains a family of groups
منابع مشابه
The Homotopy of Topological Graphs Based on Khalimsky Arcs
In this paper, we aim to develop a suitable homotopy theory of finite topological graphs by Khalimsky arcs. The notions developed are considered for the investigation of the algebraic invariants of topological graphs for their topological and graphical classifications. Keywords—Connected ordered topological spaces, digital topology, homotopy, Khalimsky spaces, topological graphs.
متن کاملHom complexes and homotopy theory in the category of graphs
We investigate a notion of ×-homotopy of graph maps that is based on the internal hom associated to the categorical product in the category of graphs. It is shown that graph ×homotopy is characterized by the topological properties of the Hom complex, a functorial way to assign a poset (and hence topological space) to a pair of graphs; Hom complexes were introduced by Lovász and further studied ...
متن کاملOn the topology of simplicial complexes related to 3-connected and Hamiltonian graphs
Using techniques from Robin Forman’s discrete Morse theory, we obtain information about the homology and homotopy type of some graph complexes. Specifically, we prove that the simplicial complex ∆n of not 3-connected graphs on n vertices is homotopy equivalent to a wedge of (n − 3) · (n − 2)!/2 spheres of dimension 2n − 4, thereby verifying a conjecture by Babson, Björner, Linusson, Shareshian,...
متن کاملHomotopy limits in type theory
Working in homotopy type theory, we provide a systematic study of homotopy limits of diagrams over graphs, formalized in the Coq proof assistant. We discuss some of the challenges posed by this approach to formalizing homotopy-theoretic material. We also compare our constructions with the more classical approach to homotopy limits via fibration categories.
متن کاملDiscrete Morse Theory and the Homotopy Type of Clique Graphs
We attach topological concepts to a simple graph by means of the simplicial complex of its complete subgraphs. Using Forman’s discrete Morse theory we show that the strong product of two graphs is homotopic to the topological product of the spaces of their complexes. As a consequence, we enlarge the class of clique divergent graphs known to be homotopy equivalent to all its iterated clique graphs.
متن کاملHomotopy equivalence of isospectral graphs
In previous work we defined a Quillen model structure, determined by cycles, on the category Gph of directed graphs. In this paper we give a complete description of the homotopy category of graphs associated to our model structure. We endow the categories of N-sets and Z-sets with related model structures, and show that their homotopy categories are Quillen equivalent to the homotopy category H...
متن کامل