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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.


This paper is supported jointly by a Nature Science Foundation of China (11301470) and a Nature Science Foundation of Yunnan Province (2012FB139).


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: 2014-05-03
Accepted: 2015-01-31
Published Online: 2017-02-28
Published in Print: 2017-03-01

© 2017 Mathematical Institute Slovak Academy of Sciences