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Licensed Unlicensed Requires Authentication Published by De Gruyter February 28, 2017

When is the cayley graph of a semigroup isomorphic to the cayley graph of a group

Shoufeng Wang
From the journal Mathematica Slovaca

Abstract

It is well known that Cayley graphs of groups are automatically vertex-transitive. A pioneer result of Kelarev and Praeger implies that Cayley graphs of semigroups can be regarded as a source of possibly new vertex-transitive graphs. In this note, we consider the following problem: Is every vertex-transitive Cayley graph of a semigroup isomorphic to a Cayley graph of a group? With the help of the results of Kelarev and Praeger, we show that the vertex-transitive, connected and undirected finite Cayley graphs of semigroups are isomorphic to Cayley graphs of groups, and all finite vertex-transitive Cayley graphs of inverse semigroups are isomorphic to Cayley graphs of groups. Furthermore, some related problems are proposed.

Acknowledgement

The author would like to thank the referees for their valuable suggestions which lead to a great improvement of this paper.

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Received: 2014-5-3
Accepted: 2015-1-31
Published Online: 2017-2-28
Published in Print: 2017-2-1

© 2017 Mathematical Institute Slovak Academy of Sciences