Complemented zero-divisor graphs associated with finite commutative semigroups

Abstract

Let S be a commutative semigroup with zero. The zero-divisor graph associated with S is the simple graph whose vertex set is given by the nonzero zero-divisors of S where two distinct vertices are adjacent if and only if their product is zero. A graph is called complemented if every vertex has an incident edge which is not the edge of any three cycle. Complemented zero-divisor graphs which are associated with a finite commutative semigroup are described in terms of their clique number. The algebraic properties of the finite semigroups S whose zero-divisor graphs are complemented are described in terms of the clique number of the graph. Infinite families of complemented zero-divisor graphs associated with a finite commutative semigroup are given.

KEYWORDS:

• Complemented graph
• commutative semigroup
• simple graph
• zero-divisor graph

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13A70
• Secondary: 20M14
• 05C25

Acknowledgments

The authors would like to thank the referee for the careful reading and insightful comments which greatly improved this paper.