Let H be an abelian group written additively and k be a positive integer. Let G(H, k) denote the digraph whose set of vertices is just H, and there exists a directed edge from a vertex a to a vertex b if b = ka. In this paper we give a necessary and sufficient condition for G(H, k1) ≃ G(H, k2). We also discuss the problem when G(H1, k) is isomorphic to G(H2, k) for a given k. Moreover, we give an explicit formula of G(H, k) when H is a p-group and gcd (p, k)=1.
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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).
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.
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