$E_k$-Super Vertex Magic Labeling of Graphs

Abstract

Let $G$ be a graph with $p$ vertices and $q$ edges. An $E_k$-super vertex magic labeling ($E_k$-SVML) is a bijection $f: V(G) \cup E(G)\rightarrow \{ 1,2 , \ldots, p + q \}$ with the property that $f(E(G)) = \{ 1,2,\ldots, q \}$ and for each $v \in V(G)$, $f(v) + w_k(v) = M$ for some positive integer $M$. For an integer $k \geq 1$ and for $v \in V(G)$, let $w_k(v) = \sum\limits_{e \in E_k(v)}f(e)$, where $E_k (v)$ is the set of all edges which are at distance at most $k$ from $v$. The graph $G$ 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 $e \in E(G)$. A graph that admits an $E_k$-SVML is called $E_k$-super vertex magic ($E_k$-SVM). This paper contain several properties of $E_k$-SVML in graphs. A necessary and sufficient condition for the existence of $E_k$-SVML in graphs has been obtained. Also, the magic constant for $E_k$-regular graphs has been obtained. Further, we establish $E_2$-SVML of some classes of graphs such as cycles, complement of cycles, prism graphs and a family of circulant graphs.