Zero-divisor semigroups and refinements of a star graph. (Q1025512)

A simple graph \(G\) is called a `refinement' of a connected simple graph \(H\) if \(V(G)=V(H)\) and \(a-b\) in \(H\) implies \(a-b\) in \(G\) for all distinct vertices of \(G\), where \(a-b\) means that \(a\neq b\) and \(a\) is adjacent to \(b\). A semigroup is called a `zero-divisor semigroup' if it has a zero element and every non-zero element is a zero-divisor. For a commutative zero-divisor semigroup \(S\), the zero-divisor graph \(\Gamma(S)\) is defined as having the nonzero zero-divisors of \(S\) as vertices and \(u-v\) an edge if and only if \(u\neq v\) and \(uv=0\). The article studies the correspondence between commutative zero-divisor semigroups and refinements of star graphs. The structure of zero-divisor semigroups \(S\) is found in a number of cases; for example, it is proved that for any (finite or infinite) cardinal number \(n\geq 2\), there exists exactly one (commutative) nilpotent semigroup \(S\) such that \(S^3\neq 0\) and \(\Gamma(S)\) is isomorphic to the star graph \(K_{1,n}\). For some types of finite graphs \(G\), the number of mutually non-isomorphic zero-divisor semigroups having \(G\) as its graph is counted.