Asian Journal of Mathematics

Volume 19 (2015)

Number 5

Cohomology of digraphs and (undirected) graphs

Pages: 887 – 932

DOI: https://dx.doi.org/10.4310/AJM.2015.v19.n5.a5

Authors

Alexander Grigor’yan (Department of Mathematics, University of Bielefeld, Germany)

Yong Lin (Department of Mathematics, Renmin University of China, Beijing, China)

Yuri Muranov (Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Olsztyn, Poland)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)

Abstract

We construct a cohomology theory on a category of finite digraphs (directed graphs), which is based on the universal calculus on the algebra of functions on the vertices of the digraph. We develop necessary algebraic technique and apply it for investigation of functorial properties of this theory. We introduce categories of digraphs and (undirected) graphs, and using natural isomorphism between the introduced category of graphs and the full subcategory of symmetric digraphs we transfer our cohomology theory to the category of graphs. Then we prove homotopy invariance of the introduced cohomology theory for undirected graphs. Thus we answer the question of Babson, Barcelo, Longueville, and Laubenbacher about existence of homotopy invariant homology theory for graphs. We establish connections with cohomology of simplicial complexes that arise naturally for some special classes of digraphs. For example, the cohomologies of posets coincide with the cohomologies of a simplicial complex associated with the poset. However, in general the digraph cohomology theory can not be reduced to simplicial cohomology. We describe the behavior of digraph cohomology groups for several topological constructions on the digraph level and prove that any given finite sequence of non-negative integers can be realized as the sequence of ranks of digraph cohomology groups. We present also sufficiently many examples that illustrate the theory.

Keywords

(co)homology of digraphs, (co)homology of graphs, differential graded algebras, path complex of a digraph, simplicial homology, differential calculi on algebras

2010 Mathematics Subject Classification

05C25, 05C38, 16E45, 18G35, 18G60, 55N35, 55U10, 57M15

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Published 20 November 2015