Note on group distance magicness on product graphs

  • Appattu Vallapil Prajeesh Department of Mathematics, National Institute of Technology Calicut, Kozhikode 673601, India.
  • Krishnan Paramasivam Department of Mathematics, National Institute of Technology Calicut, Kozhikode 673601, India.

Abstract

If $l$ is a bijection from the vertex set $V(G)$ of a graph $G$ to an additive abelian group $\Gamma$ of $|V(G)|$ elements such that for any vertex $u$ of $G$, the weight $\sum_{v\in N_{G}(u)}l(v)$ is $\mu$, where $\mu \in \Gamma$, then $l$ is a $\Gamma$-distance magic labeling of $G$. A graph $G$ that admits such an $l$ is $\Gamma$-distance magic and if $G$ is $\Gamma$-distance magic for every such $\Gamma$, then $G$ is a group distance magic graph.

In this paper, we provide some results on the group distance magicness of the lexicographic and direct product of two graphs. By proving a few necessary conditions, we characterize the group distance magicness of a tree. In addition, we find three techniques to construct group distance magic graphs recursively from the existing ones and with respect to any abelian group with one involution, we determine infinitely many nongroup distance magic graphs.

Author Biographies

Appattu Vallapil Prajeesh, Department of Mathematics, National Institute of Technology Calicut, Kozhikode 673601, India.

National Institute of Technology Calicut (NIT Calicut) - 44th Rank by The National Institutional Ranking Framework (NIRF) by the Ministry of Human Resource Development (MHRD), Goverment of India. 

Krishnan Paramasivam, Department of Mathematics, National Institute of Technology Calicut, Kozhikode 673601, India.
National Institute of Technology Calicut (NIT Calicut) - 44th Rank by The National Institutional Ranking Framework (NIRF) by the Ministry of Human Resource Development (MHRD), Goverment of India. 
Published
2021-03-19
Section
Articles