On Structural and Spectral Properties of Distance Magic Graphs

TL;DR

The paper investigates structural and spectral aspects of distance magic graphs, introducing the generalized $p$-distance magic labeling and deriving spectral criteria for distance magic via the adjacency matrix $A$ and Laplacian $L$, including a key condition that the all-ones vector is an eigenvector of the operator $APT$. It shows that distance magic implies $p$-distance magic for infinitely many $p$, and provides a concrete $(L^2 + A^2)$-based criterion for connected even-regular graphs, along with a doubly stochastic property of $AA^{\dagger}$. The automorphism group action on the labeling set yields exact divisibility relations between $|Aut(G)|$ and the number of labelings $|\mathcal{M}(G)|$, with sharp equalities for some small graphs; the work also constructs infinite families of singular distance magic graphs via cone operations. Together, these results offer spectral and group-theoretic tools for constructing, recognizing, and classifying distance magic labelings and point to rich avenues for future research in graph labeling and its algebraic underpinnings.

Abstract

A graph $G=(V,E)$ is said to be distance magic if there is a bijection $f$ from a vertex set of $G$ to the first $|V(G)|$ natural numbers such that for each vertex $v$, its weight given by $\sum_{u \in N(v)}f(u)$ is constant, where $N(v)$ is an open neighborhood of a vertex $v$. In this paper, we introduce the concept of $p$-distance magic labeling and establish the necessary and sufficient condition for a graph to be distance magic. Additionally, we introduce necessary and sufficient conditions for a connected regular graph to exhibit distance magic properties in terms of the eigenvalues of its adjacency and Laplacian matrices. Furthermore, we study the spectra of distance magic graphs, focusing on singular distance magic graphs. Also, we show that the number of distance magic labelings of a graph is, at most, the size of its automorphism group.

Figures (6)

• Figure 1: A $2$-distance magic labeling of a graph
• Figure 2: $2$-magic graphs with an Eulerian subgraph or a matching.
• Figure 3: $p$-distance magic labelings of $C_4$
• Figure 4: Non-singular distance magic graph of the smallest order
• Figure 5: Distance magic labeling of graphs $P_3$ and $C_4$

Theorems & Definitions (38)

• Definition 1.1
• Definition 1.2
• Theorem 1.1
• Theorem 1.2
• proof
• Theorem 2.1
• proof
• Corollary 2.1.1
• Corollary 2.1.2
• Theorem 2.2