E_k- Super vertex in-magic total labeling of directed graphs

Abstract

Let $D = (V, A)$ be a directed graph with $p$ vertices and $q$ arcs. For an integer $k \geq 1$ and for $v \in V(D)$, let $w_k(v) = \sum\limits_{e \in E_k(v)}f(e)$, where $E_k (v)$ is the set of all in-neighborhood arcs which are at distance at most $k$ from $v$. The digraph $D$ is said to be \textit{$E_k$-regular} with regularity $r$ if and only if $\left| E_k(e) \right| = r$ for some integer $r \geq 1$ and for all $e \in A(D)$, where $E_k(e)=E_k(u,v)=\{w \in V(D) : 1 \leq d(w,v) \leq k \}$. An $E_k$-super vertex in-magic total labeling ($E_k$-SVIMTL) is a bijection $f: V(D) \cup A(D) \rightarrow \{ 1,2 , \ldots, p + q \}$ with the property that $f(A(D)) = \{ 1,2,\ldots, q \}$ and for each $v \in V(D)$, $f(v) + w_k(v) = M$ for some positive integer $M$. A digraph that admits an $E_k$-SVIMTL is called an $E_k$-super vertex in-magic total ($E_k$-SVIMT). This paper contains several properties of $E_k$-SVIMTL in digraphs. We obtain a necessary and sufficient condition for the existence of $E_k$-SVIMTL in digraphs and the magic constant for $E_k$-regular digraphs. The $E_k$-SVIMTL of unidirectional paths and star graphs have been discussed. Finally, necessary and sufficient conditions are given for the existence of a $E_2$-SVIMTL in unidirectional cycles and union of unidirectional cycles.