Zero-divisor graph with seven vertices

TL;DR

This paper classifies all zero-divisor graphs with seven vertices and the corresponding commutative semigroups with eight elements, extending previous classifications from six vertices.

Contribution

It provides a complete classification of zero-divisor graphs with seven vertices and identifies connected graphs satisfying certain conditions but not being zero-divisor graphs.

Findings

All zero-divisor graphs with seven vertices are classified.

All commutative semigroups with eight elements corresponding to these graphs are characterized.

Some graphs meet necessary conditions but are not zero-divisor graphs.

Abstract

Inspired by the work in \cite{sauer} regarding the classification of all the zero-divisor graphs with six vertices, we obtain all the zero-divisor graphs with seven vertices. Hence we classify all the zero-divisor commutative semigroups with 8 elements. We also obtain all the connected graphs with seven vertices which satisfies the necessary condition $\star$ of zero-divisor graphs given in \cite{fl} but are not the zero-divisor graphs.