Cohomology of digraphs and (undirected) graphs

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.Key words. (co)homology of digraphs, (co)homology of graphs, differential graded algebras, path complex of a digraph, simplicial homology, differential calculi on algebras.AMS subject classifications. 05C25, 05C38, 16E45, 18G35, 18G60, 55N35, 55U10, 57M15. Introduction.In this paper we consider finite simple digraphs (directed graphs) and (undirected) simple graphs. A simple digraph G is couple (V, E) where V is any set and E ⊂ {V × V \ diag}. Elements of V are called the vertices and the elements of E -directed edges. Sometimes, to avoid misunderstanding, we shall use the extended notations V G and E G instead of V and E, respectively. The fact that (a, b) ∈ E will be denoted by a → b. A (undirected) graph G is a pair (V, E) (or more precise (V G , E G )) where V is a set of vertices and E is a set of unordered pairs (v, w) of vertices. The elements of E are called edges. In this paper we shall consider only simple graphs, which have no edges (v, v) (loops).A digraph is a particular case of a quiver. A particular example of a digraph is a poset (partially ordered set) when E is just a partial order (that is, a → b if and only if a ≥ b). The interest to construction of some type of algebraic topology on the digraphs and graphs is motivated by physical applications of this subject (see, for example, [6], [7], [8]), discrete mathematics [24], [18], [4], and graph theory [1], [2], [18], and [19, Part III].Dimakis and Müller-Hoissen suggested [7] and [8] a certain approach to construction of cohomologies on digraphs, which is based on the notion of a differential *

Path homology theory of multigraphs and quivers

“…Consider at first a quiver Q 1 ("cycle") as on the diagram: It is easy to check, that ∂ω = 0, and hence H 2 (Q, Z) = 0. Using the iteration of the suspension it is relatively easy to construct quivers with nontrivial homology group in any dimension similarly to [16]. Thus we obtain H i (Q, Z) = Z, for i = 0, Z 4 , for i = 1.…”

“…In the case of quivers of power N = 1 (digraphs) without loops the homology theory was constructed in the papers [15], [16], [17]. It is an easy exercise to transfer results of the present paper to the case of simple digraphs and to check that the obtained homology theories are isomorphic.…”

“…It is an easy exercise to transfer results of the present paper to the case of simple digraphs and to check that the obtained homology theories are isomorphic. One of advantages of the construction of the present paper is that it provides a homology theory for quivers of power N = 1 allowing loops, that contains as a particular case the theory [15], [16], [17].…”

“…The path cohomology for digraphs was introduced by Dimakis and Müller-Hoissen in [10], [4], [11]. This approach was developed in [15], [16], [17], [18], and [19], where deep relations between path homology groups, Hochschild homology, simplicial homology, and Atkins theory were obtained (see also [1], [2], [3], and [6]). The path homology theory has good functorial properties, it is compatible with the homotopy theory on the graphs (digraphs), and respects the basic graph-theoretical operations: the Cartesian product and the join of two digraphs.…”

“…In the present paper we construct a homotopy invariant homology theory for quivers and multigraphs, that is a natural generalization of the path homology theory for simple digraphs and non-directed graphs that was introduced and developed in [15], [16], [17], and [18]. Then we discuss possible applications of the results and provide several examples of computations.…”