Enumerating the distance magic labelings of a distance magic graph
Let $G = (V,E)$ be a graph of order $n$. A bijection $f : V \rightarrow \{1,2,\cdots,n\}$ is a distance magic labeling of $G$ if there exists a positive integer $k$ such that $\sum_{u \in N(v)}f(u) = k$ for all $v \in V$, where $N(v)$ is the neighborhood of $v$. Any graph which admits a distance magic labeling is called a distance magic graph. In this article, we give a partial solution to the problem by Rao et al.[10] to predict all distance magic labelings of cartesian product of two cycles, $C_m \Box C_m$, where $m\equiv 2 \mod 4$. Further, we prove that the number of distance magic labelings of a distance magic graph is a multiple $| Aut(G)|$ where $Aut(G)$ is the automorphism group of the distance magic graph $G$.
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Chespin · 2026-07-20 12:09:35 EST
Summary
The paper addresses the enumeration of distance magic labelings for distance magic graphs, focusing on the Cartesian product of two cycles $C_m \Box C_m$ where $m \equiv 2 \mod 4$. It proves that the number of such labelings is a multiple of the automorphism group size $|Aut(G)|$, offering a structural insight into the relationship between graph symmetries and labelings.
Mathematical/empirical assessment
The paper leverages group actions and permutation matrices to establish that each automorphism of a distance magic graph generates a distinct distance magic labeling. This is supported by Theorem 5, which shows that the total number of labelings is $k|Aut(G)|$, with $k$ representing the number of non-equivalent labelings. The approach is grounded in the interplay between adjacency matrices and automorphisms, as seen in Eq. (3), where $AX = \mu \bar{1}$ under a distance magic labeling.
Strengths
The work provides a clear theoretical framework linking automorphism groups to distance magic labelings, offering a novel perspective on counting such labelings. The use of group action principles and permutation matrices is both elegant and mathematically rigorous. The paper also includes concrete examples and tables that illustrate the relationship between automorphism sizes and the number of labelings.
Concerns
While the paper makes progress on partial solutions, it does not fully resolve the problem posed by Rao et al. regarding all distance magic labelings of $C_m \Box C_m$. Additionally, the generalization to arbitrary distance magic graphs remains speculative without further empirical validation. The focus on specific cases may limit broader applicability.
Final decision
Weak accept