\(V_k\)-Super vertex magic graceful labeling of graphs

Abstract

Let $G$ be a finite and simple $(p, q)$ graph. An one-one onto function $f: V(G) \cup E(G) \rightarrow\{1,2,3, \ldots, p+q\}$ is called $V$ super vertex magic graceful labeling if $f(V(G))=\{1,2,3, \ldots, p\}$ and for any vertex $v \in V(G), \sum_{u \in N(v)} f(u v)-f(v)=M$, where $M$ is a whole number. For an integer $k \geq 1$, let $E_k(v)=\{e \in E(G)$ : the distance between $e$ from $v$ is less than or equal to $k$. For $v \in V(G)$, we define $w_k(v)=\sum_{e \in E_k(v)} f(e)$. A $V_k$-super vertex magic graceful labeling $\left(V_k\right.$-SVMGL) is a one-one function $f$ from $V(G) \cup E(G)$ onto the set $\{1,2,3, \ldots, p+q\}$ such that $f(V(G))=\{1,2,3, \ldots, p\}$ and for any element $v \in V(G)$, we have $w_k(v)-f(v)=M$, where $M$ is a whole number. In this paper, we study several properties of $V_k$-SVMGL and we identify an equivalent condition for the $E_k$-regular graphs which admits $V_k$-SVMGL. At last we identify some families of graphs which admit $V_2$-SVMGL.