On closed distance magic circulants of valency up to $5$

TL;DR

This paper classifies all connected circulant graphs with valency up to 5 that admit a closed distance magic labeling, expanding understanding of such labelings in specific algebraic graph structures.

Contribution

It provides a complete classification of connected closed distance magic circulants with valency at most 5, a previously unexplored case in graph labelings.

Findings

Identifies all such circulants with valency up to 5

Establishes conditions for closed distance magic labelings in these graphs

Enhances understanding of algebraic graph labelings and their classifications

Abstract

Let $\Gamma=(V,E)$ be a graph of order $n$. A {\em closed distance magic labeling} of $\Gamma$ is a bijection $\ell : V \to \{1,2, \ldots, n\}$ for which there exists a positive integer $r$ such that $\sum_{x \in N[u]} \ell(x) = r$ for all vertices $u \in V$, where $N[u]$ is the closed neighborhood of $u$. A graph is said to be {\em closed distance magic} if it admits a closed distance magic labeling. In this paper, we classify all connected closed distance magic circulants with valency at most $5$, that is, Cayley graphs $\operatorname{Cay}(\mathbb{Z}_n;S)$ where $|S| \le 5$ and $S$ generates $\mathbb{Z}_n$.