Symmetric Digraphs from Powers Modulo <i>n</i>

Abstract

For each pair of positive integers n and k, let G(n,k) denote the digraph whose set of vertices is H = {0,1,2,•••, n-1} and there is a directed edge from a  H to b  H if a k  b(mod n). The digraph G(n,k) is symmetric if its connected components can be partitioned into isomorphic pairs. In this paper we obtain all symmetric G (n,k).

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

“…Symmetric digraphs G(n, k) have been characterized completely in [5] and [14]. The following results show the relationship between the symmetric property and the semiregularity property of G(n, k).…”

“…The digraphs G(n, k) were first studied extensively by L. Somer and M. Křížek in [13], [15], [12] and [14], and also later by other authors in [19], [5], and [7], based on ideas of S. Bryant [1], G. Chassé [3], T.D. Rogers [10], and L. Szalay [16].…”

“…Amplify Sawkmie -Madan Mohan Singh Lemma 2.12 ( [5]). Let λ(p e ) = uv, where u is the largest factor of λ(p e ) relatively prime to k. Then h(T (p e , k, 1)) = min{i : v|k i }.…”

The Structure of Isomorphic Digraph from Powers Modulo pe

“…Křížek obtained all ( , ) which are symmetric of order 2. Deng and Yuan [12] determined all ( , ) which are symmetric of order , where has an odd prime divisor. Meemark and Wiroonsri [13,14] investigated the structure of ( , ), where is the quotient ring of polynomials over finite fields.…”

On the tree structure of the power digraphs modulo n

“…In Section 2, we give some basic properties of the Carmichael lambda function. In Section 3, we state basic results on G(n, k) proved in [1], [2], [4]- [7]. In Section 4, we discuss important properties of the fundamental constituents of G(n, k), which will be used throughout the paper.…”