A Characterization for $V_4$-Vertex Magicness of Trees with Diameter 5

Abstract

Let G be an undirected simple graph with vertex set V(G) and the edge set E(G) and $\mathcal {A}$ be an additive Abelian group with the identity element 0. A function $l:V(G)\rightarrow \mathcal {A}\setminus \{0\}$ is said to be a $\mathcal {A}$-vertex magic labeling of G if there exists an element $\mu$ of $\mathcal {A}$ such that $w(v)=\sum _{u\in N(v)}l(u)=\mu$ for any vertex v of G. A graph G having $\mathcal {A}$-vertex magic labeling is called a $\mathcal {A}$-vertex magic graph. If G is $\mathcal {A}$-vertex magic graph for every non-trivial additive Abelian group $\mathcal {A}$, then G is called a group vertex magic graph. In this article, a characterization for the $\mathcal {A}$-vertex magicness of any tree T with diameter 5, is given, when $\mathcal {A}\cong \mathbb {Z}_2 \oplus \mathbb {Z}_2$.