Displaying similar documents to “Graph cycles and diagram commutativity”

Travel groupoids

Ladislav Nebeský (2006)

Czechoslovak Mathematical Journal

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In this paper, by a travel groupoid is meant an ordered pair ( V , * ) such that V is a nonempty set and * is a binary operation on V satisfying the following two conditions for all u , v ∈ V : ( u * v ) * u = u ; if ( u * v ) * v = u , then u = v . Let ( V , * ) be a travel groupoid. It is easy to show that if x , y ∈ V , then x * y = y if and only if y * x = x . We say that ( V , * ) is on a (finite or infinite) graph G if V ( G ) = V and E ( G ) = { { u , v } u , v ∈ V and u ≠ u * v = v } . Clearly, every travel groupoid is on exactly one graph. In this paper, some properties of travel groupoids on graphs are studied.

Travel groupoids on infinite graphs

Jung Rae Cho, Jeongmi Park, Yoshio Sano (2014)

Czechoslovak Mathematical Journal

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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. We can associate a graph with a travel groupoid. 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 of finite graphs having a travel groupoid. In this paper, we study travel groupoids on infinite...

Matrices induced by arithmetic functions, primes and groupoid actions of directed graphs

Ilwoo Cho, Palle E. T. Jorgensen (2015)

Special Matrices

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In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By defining an injective map α from the graph groupoid G of a directed graph G to the algebra A of all arithmetic functions, we establish a corresponding subalgebra AG = C*[α(G)]︀ of A. We construct a suitable representation of AG, determined both by G and by an arbitrarily fixed prime p. And then based on this representation,...