Graph homotopy and Graham homotopy

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.

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“…These, however, do not have to match the higher dimensional substructures of G that are predetermined by G (see, for example, [1] and [7]). …”

Section: Introductionmentioning

confidence: 99%

Graphs associated with simplicial complexes

Grigor’yan

Muranov

Yau

2014

Homology, Homotopy and Applications

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“…This alternative view of the hyperbolicity constant is an interesting subject since the tractability of a problem in many applications is related to the tree-like degree of the space under investigation. (see, e.g., [58]). Furthermore, it is well known that any Gromov hyperbolic space with n points embeds into a tree metric with distortion O(δ log n) (see, e.g., [3], p. 33).…”

Section: Definitions and Backgroundmentioning

confidence: 99%

Gromov Hyperbolicity in Mycielskian Graphs

et al. 2017

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Since the characterization of Gromov hyperbolic graphs seems a too ambitious task, there are many papers studying the hyperbolicity of several classes of graphs. In this paper, it is proven that every Mycielskian graph G M is hyperbolic and that δ(G M ) is comparable to diam(G M ). Furthermore, we study the extremal problems of finding the smallest and largest hyperbolicity constants of such graphs; in fact, it is shown that 5/4 ≤ δ(G M ) ≤ 5/2. Graphs G whose Mycielskian have hyperbolicity constant 5/4 or 5/2 are characterized. The hyperbolicity constants of the Mycielskian of path, cycle, complete and complete bipartite graphs are calculated explicitly. Finally, information on δ(G) just in terms of δ(G M ) is obtained.

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“…In fact, the hyperbolicity constant of a geodesic metric space can be viewed as a measure of how "tree-like" the space is, since those spaces X with δ(X) = 0 are precisely the metric trees. This is an interesting subject since, in many applications, one finds that the borderline between tractable and intractable cases may be the tree-like degree of the structure to be dealt with (see, e.g., [16]). However, the hyperbolicity constant does not relate the graph in question to a specific tree (if connected) or forest (if not connected).…”

Section: Introductionmentioning

confidence: 99%