Snakes Related Sum Perfect Square Graphs

Abstract

Let $G = (V, E)$ be a simple $(p,q)$ graph and $f:V(G)\rightarrow\{0,1,2,\ldots, p-1\}$ be a bijection. We define $f^{\ast}:E(G)\rightarrow\ \mathbb{N}$ by $f^{\ast}(uv)=(f(u))^{2} + (f(v))^{2} + 2 f(u) \cdot f(v)$, $\forall uv \in E(G)$. If $f^{\ast}$ is injective, then $f$ is called sum perfect square labeling. A graph which admits sum perfect square labeling is called sum perfect square graph. In this paper we prove that several snakes related graphs are sum perfect square.