Edge pair sum labeling of butterfly graph with shell order

Abstract

The concept of an edge pair sum labeling was introduced in [3]. Let $G(p, q)$ be a graph. An injective map $f: E(G) \rightarrow\{ \pm 1, \pm 2, \cdots, \pm q\}$ is said to be an edge pair sum labeling if the induced vertex function $f^*: V(G) \rightarrow Z-\{0\}$ defined by $f^*(v)=\sum_{e \epsilon E_v} f(e)$ is one- one where $E_v$ denotes the set of edges in $G$ that are incident with a vertex $v$ and $f^*(V(G))$ is either of the form $\left\{ \pm k_1, \pm k_2, \cdots, \pm k_{\frac{p}{2}}\right\}$ or $\left\{ \pm k_1, \pm k_2, \cdots, \pm k_{\frac{p-1}{2}}\right\}$ $\cup\left\{ \pm k_{\frac{p+1}{2}}\right\}$ according as $p$ is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that the shell graph and butterfly graph with shell order are edge pair sum graphs.