New results on edge pair sum graphs

Abstract

Let $G$ be a (p,q) graph. An injective map $f:E(G)\rightarrow\left\{\pm1,\pm2,\cdots,\pm q \right\}$ is said to be an edge pair sum labeling if the induced vertex function $f^{*}:V(G)\rightarrow Z-\left\{0\right\}$ defined by $f^{*}(v)= \mathop\sum\limits_{e\epsilon E_{v}}f\left(e\right)$ 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 $\left\{\pm k_{1},\pm k_{2},\cdots,\pm k_{\frac{p-1}{2}}\right\}$ $\bigcup$$\left\{\pm k_{\frac{p+1}{2}}\right\}$ according as $p$ is even or odd. A graph that admits an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that the graphs jelly fish, Y-tree, theta, the subdivision of spokes in wheel $SS(W_{n})$, $P_{m} + 2K_{1}$, $C_{4} \times P_{m}$, $P_{n}\odot K^{c}_{m}$ admit edge pair sum labeling.