Fundamental groupoids of k-graphs

Fundamental Groupoids for Graphs

In this paper, we develop a $\times$-homotopy fundamental groupoid for graphs, and show a functorial relationship to the 2-category of graphs. We further explore the fundamental groupoid of graph products and develop a groupoid product which respects the graph product. A van Kampen Theorem for these groupoids is provided. Finally, we generalize previous work on a fundamental group for graphs, developing a looped walk groupoid and showing a connection to the polyhedral complex of graph morphisms.

完全多部图上的 travel groupoid

Travel groupoids on complete multipartite graphs

本文研究有限完全多部图上的 travel groupoid，完成其结构刻画并给出相应数量。

A construction of smooth travel groupoids on finite graphs

A travel groupoid is an algebraic system related with graphs. In this paper, we give an algorithm to construct smooth travel groupoids for any finite graph. This algorithm gives an answer of L.~Nebesk$\acute{\mbox{y}}$'s question, "Does there exists a connected graph $G$ such that $G$ has no smooth travel groupoid?", in finite cases.

T-partition systems and travel groupoids on a graph

The notion of travel groupoids was introduced by L. Nebeský in 2006 in connection with a study on geodetic graphs. A travel groupoid is a pair of a set $V$ and a binary operation $*$ on $V$ satisfying two axioms. For a travel groupoid, we can associate a graph. We say that a graph $G$ has a travel groupoid if the graph associated with the travel groupoid is equal to $G$. Nebeský gave a characterization for finite graphs to have a travel groupoid. In this paper, we introduce the notion of T-partition systems on a graph and give a characterization of travel groupoids on a graph in terms of T-par…

Isotopy Quotients of Hopf Algebroids and the Fundamental Groupoid of Digraphs

We build on our construction of Hopf algebroids from noncommutative calculi under the further assumption of surjectivity for the calculus. We also introduce the notions of Hopf ideals and isotopy quotients for arbitrary Hopf algebroids. Using these ingredients, we prove a Riemann-Hilbert correspondence for digraphs, by showing that the groupoid algebra of the fundamental groupoid of a digraph is isomorphic to the isotopy quotient of the Hopf algberoid corresponding to flat connections over the digraph.

Complete graph decompositions and p-groupoids

We study P-groupoids that arise from certain decompositions of complete graphs. We show that left distributive P-groupoids are distributive, quasigroups. We characterize P-groupoids when the corresponding decomposition is a Hamiltonian decomposition for complete graphs of odd, prime order. We also study a specific example of a P-quasigroup constructed from cyclic groups of odd order. We show such P-quasigroups have characteristic left and right multiplication groups, as well as the right multiplication group is isomorphic to the dihedral group.