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

Guoce Xin, Yueming Zhong, Yangbiao Zhou

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 Bóna et al.\cite{Bona-1,Bona}.

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

Abstract

Stanley's theorem establishes that for any finite graph , the number of magic labelings with magic sum can be expressed as a sum of two polynomials in . However, determining the precise form of is generally challenging. This paper aims to compute and its generating function for pseudo-line graphs and pseudo-cycle graphs, thereby extending the earlier work of Bóna et al.\cite{Bona-1,Bona}.
Paper Structure (15 sections, 32 theorems, 109 equations, 3 figures, 1 table)

This paper contains 15 sections, 32 theorems, 109 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Definitions
  3. Main Results
  4. Structure
  5. Proof of Theorems \ref{['theo-1']} and \ref{['theo-2']}
  6. Transfer matrix method
  7. Establishing Relations for the Characteristic Polynomials of $A_n$, $B_n$, $B_n^{-1}$ and $D_n$
  8. Recurrence Relations for $\bar{f}_{B_n}(y)$, $g_n(y)$ and Proofs of Main Theorems
  9. Proof of Theorem \ref{['theo-hck']}
  10. Magic Labellings for Pseudo-Cycle Graphs $C_{n,\mathbf{k}}$
  11. Vertices of the Polytope $\mathcal{P}(C_{n,1})$
  12. Polyhedral Decomposition and Explicit Expression of the Generating Function
  13. Proof of Theorem \ref{['theo-hck']}
  14. MacMahon's Partition Analysis
  15. Generating Functions of $h_{C_{n,2}}(s)$ and $h_{L_{n,2}}(s)$

Key Result

Theorem 1.1

For any finite graph $G$, there exist polynomials $\varphi_G(s)$ and $\psi_G(s)$ such that: If the loop-free subgraph of $G$ is bipartite, then $\psi_G(s) = 0$, making $h_G(s)$ polynomial.

Figures (3)

  • Figure 1: The pseudo-line graph $L_{n,m}$.
  • Figure 2: The pseudo-cycle graph $C_{n,m}$.
  • Figure 3: The pseudo-cycle graphs $C_{1,m}$ and $C_{2,m}$.

Theorems & Definitions (62)

  • Theorem 1.1: Stanley, Stanley-magiclabelings
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Example 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • ...and 52 more