Vk- Super Vertex In-Magic Labelings of Directed Graphs

Let D be a digraph of order p and size q. For an integer k ⩾ 1 and ν ∈ V(D), let \(\sum _{e\in {E}_{k}(v)}f(e)\) where Ek (v) is the set of all in-arcs which are at distance at most k from ν. A V k -super vertex in-magic labeling (Vk -SVIML) 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 for every ν ∈ V(D), f(ν) + ω k(ν) = M for some positive integer M. A digraph that admits a V k -SVIML is called V k -super vertex in-magic (V k -SVIM). In this paper, we study some properties of V k -SVIML in digraphs. We characterized the digraphs which are V k -SVIM. Also, we find the magic constant for Ek -regular digraphs. Farther, we characterized the unidirectional cycles and union of unidirectional cycles which are V 2-SMM.