A groupoid should be thought of as a group with many objects, or with many identities. A precise definition is given below. A groupoid with one object is essentially just a group. So the notion of groupoid is an extension of that of groups. It gives an additional convenience, flexibility and range of applications, so that even for purely group-theoretical work, it can be useful to take a path t...

Let G=(V,E) be a simple connected graph of order p and size q. A decomposition of a graph G is a collection π of edge-disjoint subgraphs G_1,G_2,…,G_n of G such that every edge of G belongs to exactly one G_i,(1≤i ≤n). The decomposition 〖π={G〗_1,G_2,…,G_n} of a connected graph G is said to be a distinct edge geodetic decomposition if g_1 (G_i )≠g_1 (G_j ),(1≤i≠j≤n). The maximum cardinality of π...

Abstract: This paper proposes a novel representation of decomposable graphs based on semi-latent tree-dependent bipartite graphs. The novel representation has two main benefits. First, it enables a form of subclustering within maximal cliques of the graph, adding informational richness to the general use of decomposable graphs that could be harnessed in applications with behavioural type of dat...

We show that relations in Homflypt type skein theory of an oriented [Formula: see text]-manifold text] are induced from a text]-groupoid defined the fundamental space singular links text]. The module by homomorphisms related to string topology. They appear representation groupoid into free modules on set model objects. construction is singularity stratification and relates Vassiliev theory. Sev...

The purpose of this paper is to prove two theorems which concern the position of étale groupoids among general smooth (or ”Lie”) groupoids. Our motivation comes from the non-commutative geometry and algebraic topology concerning leaf spaces of foliations. Here, one is concerned with invariants of the holonomy groupoid of a foliation [4, 34], such as the cohomology of its classifying space [14],...