Spectra of Group Vertex Magic Graphs

TL;DR

This paper explores the spectral properties of group vertex magic graphs, introducing new methods and conditions for their spectra, especially for specific Abelian groups, and investigates the relationship between the spectrum and reduced spectrum.

Contribution

It presents a new approach using minimal vertices, provides necessary and sufficient conditions for spectra to form subgroups for certain groups, and introduces the concept of reduced spectrum.

Findings

Spectra form subgroups for A=V4 or Zp under certain conditions.

A new method based on minimal vertices for analyzing group vertex magic graphs.

Relationship established between spectrum and reduced spectrum.

Abstract

Let G be a simple undirected graph and let A be an additive Abelian group with identity 0. In this paper, we introduce the concept of group magic spectrum of a graph G with respect to a given Abelian group A and is defined as spec(G, A):= {{\lambda} : {\lambda} is a magic constant of some A-vertex magic labeling f }. In their recent work, K. M. Sabeel et al. in Australas. J. Combin. 85(1) (2023), 49-60 proved a forbidden subgraph characterization for the group vertex magic graph. In this work, we present a new method which uses minimum number of vertices required for this graph. We obtain a necessary and sufficient condition for the spectrum of a graph G to be a subgroup when A = V4 or Zp, where p is a prime number. Also we introduce the notion of reduced spectrum redspec(G, A) and study the relation between spec(G, A) and redspec(G, A).