Five-point zero-divisor graphs determined by equivalence classes

Abstract

We study condensed zero-divisor graphs (those whose vertices are equivalence classes of zero-divisors of a ring $R$) having exactly five vertices. In particular, we determine which graphs with exactly five vertices can be realized as the condensed zero-divisor graph of a ring. We provide the rings for the graphs which are possible, and prove that the rest of graphs can not be realized via any commutative ring. There are 34 graphs in total which contain exactly five vertices.