In this paper, we extend the results of DeMeyer and DeMeyer (J Algebra 283:190-198, 2005) that determine a set of sufficient conditions for a given graph to be the zero-divisor graph of a commutative semigroup, and use these to construct a larger set of graphs of this type. We then use these results to classify the connected graphs on six vertices, determining whether or not each is the zero-divisor graph of a commutative semigroup of order seven. To accomplish this, we give specific examples of graphs that can be easily classified using the extensions. For those graphs to which neither the original results nor the extensions apply, we provide a method by which we can classify them.
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