The Strongly Antimagic labelings of Double Spiders

A graph $G=(V,E)$ is strongly antimagic, if there is a bijective mapping $f: E \to \{1,2,\ldots,|E|\}$ such that for any two vertices $u\neq v$, not only $\sum_{e \in E(u)}f(e) \ne \sum_{e\in E(v)}f(e)$ and also $\sum_{e \in E(u)}f(e) < \sum_{e\in E(v)}f(e)$ whenever $\deg(u)< \deg(v) $, where $E(u)$ is the set of edges incident to $u$. In this paper, we prove that double spiders, the trees contains exactly two vertices of degree at least 3, are strongly antimagic.