Semiregularity of the Cubic Mapping Graphs of Quotient Rings of the Imaginary Quadratic Rings
Let K =Q(√d),where Q is the rational number field and d＜0 is a square-free integer.Let Γ(γ) be the cubic mapping graph of Rd/<γ>,where γ is a non-invertible element in Rd,the ring of algebraic integers of K.The vertex set of Γ(γ) consists of all elements of Rd/<γ>.If α3 =β,then there is a directed edge from α toβ.In this paper,we investigate the semiregularity of Γ(γ) for d =-3,-7,-11,-19,-43,-67,-163.
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