$V_k$-Super vertex magic labeling of graphs

Abstract

Let $G$ be a simple graph with $p$ vertices and $q$ edges. A $V$-super vertex magic labeling is a bijection $f: V(G) \cup$ $E(G) \rightarrow\{1,2, \ldots, p+q\}$ such that $f(V(G))=\{1,2, \ldots, p\}$ and for each vertex $v \in V(G), f(v)+\sum_{u \in N(v)} f(u v)=M$ for some positive integer $M$. A $V_k$-super vertex magic labeling $\left(V_k\right.$-SVML) is a bijection $f: V(G) \cup E(G) \rightarrow$ $\{1,2, \ldots, p+q\}$ with the property that $f(V(G))=\{1,2, \ldots, p\}$ and for each $v \in V(G), f(v)+w_k(v)=M$ for some positive integer $M$. A graph that admits a $V_k$-SVML is called $V_k$-super vertex magic. This paper contains several properties of $V_k$-SVML in graphs. A necessary and sufficient condition for the existence of $V_k$-SVML in graphs has been obtained. Also, the magic constant for $E_k$-regular graphs has been obtained. Further, we study some classes of graphs such as cycles, complement of cycles, prism graphs and a family of circulant graphs which admit $V_2$-SVML.