Vk- Super Vertex Out-Magic Labeling of Digraphs

Abstract

Let D(V, A) be a digraph of order p and size q. For an integer k ≥ 1 and for v ∈ V (D), let wk(v) = P e∈Ek(v) f(e), where Ek(v) is the set containing all arcs which are at distance at most k from v. The digraph D is said to be Ek-regular with regularity r if and only if |Ek(e)| = r for some integer r ≥ 1 and for all e ∈ A(D). A Vk-super vertex out-magic labeling (Vk-SVOML) is an one-to-one onto function f : V (D)∪A(D) → {1, 2, . . . , p+q} such that f(V (D)) = {1, 2, . . . , p} and there exists a positive integer M such that f(v) + wk(v) = M, ∀ v ∈ V (D). A digraph that admits a Vk-SVOML is called Vk-super vertex out-magic (Vk-SVOM). This paper contains several properties of Vk-SVOML in digraphs. We characterized the digraphs which are VkSVOM. Also, the magic constant for Ek-regular graphs has been obtained. Further, we characterized the unidirectional cycles and union of unidirectional cycles which are V2-SVOM