Generalized Cayley graphs over polygroups

Abstract

Polygroups are a generalization of groups in which the composition of any two elements are a non-empty set. In this paper, first we recall the concept of polygroups and introduce a new construction for building a polygroup from a polygroup and a non-empty set. Then we study the concept of generalized Cayley graphs over polygroups, say GCP-graphs. Then we prove some properties of them in order to answer this question: which simple graphs are GCP-graphs? Finally, we prove that every simple graph of order at most five is a GCP-graph.

Keywords:

• Caylay graph
• GCP-graph
• group
• polygroup
• simple graph

2010 Mathematics Subject Classification:

• 20N20
• 05C25

Acknowledgments

The authors are highly grateful to referees for their valuable comments and suggestions for improving the paper.