On the symmetric digraphs from the kth power mapping on finite commutative rings

Abstract

For a finite commutative ring R and a positive integer k, let G(R, k) denote the digraph whose set of vertices is R and for which there is a directed edge from a to ak. The digraph G(R, k) is called symmetric of order M if its set of connected components can be partitioned into subsets of size M with each subset containing M isomorphic components. We primarily aim to factor G(R, k) into the product of its subdigraphs. If the characteristic of R is a prime p, we obtain several sufficient conditions for G(R, k) to be symmetric of order M.

Digraphs associated with the kth power map on the quotient ring of polynomials over finite fields

“…The power digraphs associated with the kth power map on the quotient ring of polynomials over finite fields were first studied by Meemark and Wiroonsri [4], and briefly by Deng and Somer [2]. A similar class of digraphs, associated with the kth power map defined on the ring of integers modulo n, was studied by Somer and Křížek and others (see [6; 7] and their references).…”

“…Fq[x] f * . An explicit formula for λ(P e ) was established in [2], and it is given by λ(P e ) = p s (|P | − 1), where p s−1 < e ≤ p s . (1.1)…”

Exponent of local ring extensions of Galois rings and digraphs of the $k$th power mapping

“…This motivated Dang and Somer [2] to compute without the explicit structure of unit group the exponent of the quotient ring…”

“…Besides the characteristic of the unit group, the exponent of the ring can be used to study the digraph of the k th power mapping [2,[7][8][9]. This motivated Dang and Somer [2] to compute without the explicit structure of unit group the exponent of the quotient ring F q [x]/(f (x) a ) , where a ≥ 1, F q is the field of q elements and f (x) is a monic irreducible polynomial over…”

“…Therefore, the number of components of this digraph is equal to the number of its cycles. This functional digraph is defined using the idea of Somer and Křížek [4], who studied the structure of digraphs G (2) (Z n ) . Later, they worked on the k th power mapping digraph G (k) (Z n ) [5].…”