The Square Mapping Graphs of Finite Commutative Rings

Abstract

For a finite commutative ring R, the square mapping graph of R is a directed graph Γ(R) whose set of vertices is all the elements of R and for which there is a directed edge from a to b if and only if a 2 = b. We establish necessary and sufficient conditions for the existence of isolated fixed points, and the cycles with length greater than 1 in Γ(R). We also examine when the induced subgraph on the set of zero-divisors of a local ring with odd characteristic is semiregular. Moreover, we completely determine the finite commutative rings whose square mapping graphs have exactly two, three or four components.

The iteration digraphs of finite commutative rings

“…After this introduction, we obtain some results in Section 2 on cycles and components of G(R, k) for finite commutative rings R . These results generalize the work [15] on the digraph associated to the square mapping. In Section 3, we employ the digraphs products to explore the symmetric digraphs and obtain results parallel to those of Somer and Křížek [10].…”

“…In recent years, there has been growing interest in the iteration digraphs associated with the ring Z n of integers modulo n, the quotient ring of polynomials over finite fields, and the ring of Gaussian integers modulo n , etc. (e.g., see [1,3,4,11,13,14,15]). …”

The fundamental constituents of iteration digraphs of finite commutative rings

“…Moreover, by a proof similar to that of [7,Theorem 3.2], we derive that G 1 (R, k) is semiregular for any finite commutative ring R, so indeg Rj (a j ) = indeg Rj (b j ) for R j ∈ N . Thus = (a 1 , . .…”

On the uniqueness of the factorization of power digraphs modulo $n$

“…Rogers [10], and L. Szalay [16]. There have also been extensions of these ideas to more general structures (see [6], [8], [9], [17], and [18]).…”