Graphs associated with simplicial complexes

Abstract

The cohomology of digraphs was introduced for the first time by Dimakis and Müller-Hoissen. Their algebraic definition is based on a differential calculus on an algebra of functions on the set of vertices with relations that follow naturally from the structure of the set of edges. A dual notion of homology of digraphs, based on the notion of path complex, was introduced by the authors, and the first methods for computing the (co)homology groups were developed. The interest in homology on digraphs is motivated by physical applications and relations between algebraic and geometrical properties of quivers. The digraph G B of the partially ordered set B S of simplexes of a simplicial complex S has graph homology that is isomorphic to the simplicial homology of S. In this paper, we introduce the concept of cubical digraphs and describe their homology properties. In particular, we define a cubical subgraph G S of G B , whose homologies are isomorphic to the simplicial homologies of S.

On the path homology theory of digraphs and Eilenberg–Steenrod axioms

“…The homology and homotopy theory of digraphs considered in this paper were introduced in [16,17,18,19]. Our approach is closely related to geometric and algebraic topology [14,13,29,24,21], to physical applications of graph theory [32,11,10,4,9], to Atkins homotopy theory [1,2,3,6], to the theory of quivers [8,22,15,19], and to various discrete (co)homology and homotopy theories [5,3,6,31,12,28,23,7].…”

“…The paper is organized as follows. In Section 2, we give some preliminary material concerning the homotopy theory for digraphs (see also [16,18]) and prove several technical results.…”

Analysis and stochastic processes on metric measure spaces

“…In a series of papers [31], [32], [33], [34], we introduced the notion of a differential form on a digraph (=directed graph) with the exterior derivative d, as well as the dual object -a ∂-invariant path with the boundary operator ∂, which leads to the dual notions of cohomology and homology of graphs.…”

Fundamental groupoids of digraphs and graphs

“…The notion of homology groups H p (G, Z) of digraphs was introduced in [10] (see also [6], [7], [9]). The physical applications of homology (cohomology) theory of digraphs requires development of effective methods of computing of these groups.…”

“…In this paper we develop further the homotopy theory for digraphs (= directed graphs) initiated in [9], [8], and [10]. In the category of digraphs, the homology and the homotopy theories were introduced in [8] in such a way that the homology groups are homotopy invariant and the first homology group of a connected digraph is isomorphic to the abelization of its fundamental group.…”