On Distance Magic Harary Graphs

This paper establishes two techniques to construct larger distance magic and (a, d)-distance antimagic graphs using Harary graphs and provides a solution to the existence of distance magicness of legicographic product and direct product of G with C4, for every non-regular distance magic graph G with maximum degree |V(G)|-1.Comment: 12 pages, 1 figur

Note on group distance magic graphs G[C4]G[C_4]

A \emph{group distance magic labeling} or a \gr-distance magic labeling of a graph $G(V,E)$ with $|V | = n$ is an injection $f$ from $V$ to an Abelian group \gr of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}f(y)$ of every vertex $x \in V$ is equal to the same element \mu \in \gr, called the magic constant. In this paper we will show that if $G$ is a graph of order $n=2^{p}(2k+1)$ for some natural numbers $p$, $k$ such that \deg(v)\equiv c \imod {2^{p+1}} for some constant $c$ for any $v\in V(G)$, then there exists an \gr-distance magic labeling for any Abelian group \gr for the graph $G[C_4]$. Moreover we prove that if \gr is an arbitrary Abelian group of order $4n$ such that \gr \cong \zet_2 \times\zet_2 \times \gA for some Abelian group \gA of order $n$, then exists a \gr-distance magic labeling for any graph $G[C_4]$