$V_k$-Super vertex magic labeling of graphs

$V_k$-Super vertex magic labeling of graphs

**Authors : **

Sivagnanam Mutharasu ^{1} and Duraisamy Kumar^{ 2 *}

**Author Address : **

^{1,2} Department of Mathematics, C. B. M. College, Coimbatore - 641 042, Tamil Nadu, India.

*Corresponding author.

**Abstract : **

Let $G$ be a simple graph with $p$ vertices and $q$ edges. A $V$-super vertex magic labeling is a bijection $f: V(G) cup E(G) ightarrow { 1,2 , ldots, p + q }$ such that $f(V(G))={1,2,ldots,p}$ and for each vertex $v in V(G)$, $f(v) + sumlimits_{u in N(v)} f(uv) = M$ for some positive integer $M$. A $V_k$-super vertex magic labeling ($V_k$-SVML) is a bijection $f: V(G) cup E(G) ightarrow { 1,2 , ldots, p + q }$ 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.

**Keywords : **

Vertex magic total labeling, super vertex magic total labeling, $V_k$-super vertex magic labeling, $E_k$-regular graphs, circulant graphs.

**DOI : **

**Article Info : **

*Received : * August 14, 2018; *Accepted : * October 19, 2018.