Pair Sum Labeling of Splitting, Shadow and Middle Graphs of Path Graphs

Abstract

Let $G$ be a finite, undirected and simple graph with $p$ vertices and $q$ edges. A \textit{pair sum labeling} of $G$ is an injective map $f: V(G) \rightarrow \{\pm1, \pm2, \ldots, \pm p \}$ such that the induced edge function, $f_e: E(G) \rightarrow \mathbb{Z}-\{0\}$ defined by $f_e(uv)=f(u)+f(v)$ is one-to-one and $f_e(E(G))$ is either of the form $\left\{\pm k_1, \pm k_2, \ldots, \pm k_{\frac{q}{2}} \right\}$ or $\left\{\pm k_1, \pm k_2, \ldots, \pm k_{\frac{q-1}{2}}\right\} \cup \left\{k_{\frac{q+1}{2}}\right\}$ according as $q$ is even or odd number. A graph with pair sum labeling will be referred to as a pair sum graph. Here, we show that the splitting graph, shadow graph, and middle graph of a path graph are pair sum graphs.