Magic labelling enumeration on pseudo-line graphs and pseudo-cycle graphs

TL;DR

This paper computes the magic labeling enumeration function and its generating function for pseudo-line and pseudo-cycle graphs, extending previous work and providing explicit formulas for these classes.

Contribution

It introduces methods to determine the magic labeling count polynomial and generating function specifically for pseudo-line and pseudo-cycle graphs.

Findings

Derived explicit formulas for $h_G(s)$ for pseudo-line graphs.

Extended enumeration techniques to pseudo-cycle graphs.

Provided generating functions for these graph classes.

Abstract

Stanley's theorem establishes that for any finite graph $G$, the number $h_G(s)$ of magic labelings with magic sum $s$ can be expressed as a sum of two polynomials in $s$. However, determining the precise form of $h_G(s)$ is generally challenging. This paper aims to compute $h_G(s)$ and its generating function for pseudo-line graphs and pseudo-cycle graphs, thereby extending the earlier work of B\'{o}na et al.\cite{Bona-1,Bona}.