$m$-Magic Labeling on Anti Fuzzy Path Graph

TL;DR

This paper explores the concept of $m$-magic labelings on anti fuzzy path graphs and bipolar variants, establishing conditions for their existence based on the number of vertices and the parameter $m$.

Contribution

It introduces and analyzes $m$-magic labelings on anti fuzzy path and bipolar graphs, extending previous fuzzy graph labeling concepts with new existence conditions.

Findings

Existence of $m$-magic anti fuzzy path labelings for all natural numbers $n$ with specific bounds.

Conditions for $m$-magic anti fuzzy bipolar graph labelings based on parity and modular arithmetic.

Extension of previous fuzzy graph labeling studies to new anti fuzzy and bipolar graph classes.

Abstract

The topic of $m$-magic labeling is discussed in this article. The application of these problem is applied to path anti fuzzy and path anti fuzzy bipolar. Anti fuzzy and anti fuzzy bipolar graphs are new concepts of fuzzy graphs. This article to discover m magic anti fuzzy and anti fuzzy bipolar graphs through the adaptation of some previous studies on labeling magic, bimagic, trimagic anti fuzzy graphs, and anti fuzzy bipolar graphs. The graph path is the subject of $m$-magic anti fuzzy and anti fuzzy bipolar graph study. The results of this study show that, for every natural number $n$ dan $n \ge (2m+1+ma)$ with $a = \{0,1,2,3,\ldots,1\}$, there is an anti fuzzy $m$-magic graph path labeling. While in $m > 2$ in terms of 2 cases. Case 1 when for odd $n$ with $n \equiv 2m+1$ mod$m$ and for even $n$ with $n \equiv 2m+1$ mod$2m$, case 2 when $n \equiv m+1$ mod $2m$ for $m$ odd and $n \ne m+1$ there is $m$-magic graph path anti fuzzy bipolar labeling.