The Inter-magic Spectra of Trees

TL;DR

This paper determines the integer-magic spectra of trees with diameter six or more, extending previous results limited to trees of smaller diameter, and explores the labeling properties related to graph theory.

Contribution

It introduces the complete characterization of the integer-magic spectra for trees of diameter six and higher, advancing the understanding of graph labelings.

Findings

Complete spectra for trees of diameter six and higher are established.

Extends previous results limited to trees of diameter at most five.

Provides new insights into graph labelings and their properties.

Abstract

For any positive integer $h$, a graph $G=(V,E)$ is said to be $h$-magic if there exists a labeling $l:E(G)\to \mathbb{Z}_h -\{0\} $ such that the induced vertex set labeling $\ l^+ : V(G) \to \mathbb{Z}_h \ $ defined by $$ l^+ (v)=\sum_{uv \in E(G)} \ l(uv) $$ is a constant map. The integer-magic spectrum of a graph $G$, denoted by $IM(G)$, is the set of all $h \in \mathbb{N} $ for which $G$ is $h$-magic. So far, only the integer-magic spectra of trees of diameter at most five have been determined. In this paper, we determine the integer-magic spectra of trees of diameter six and higher.