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Characterizing the Degree-Kirchhoff, Gutman, and Schultz Indices in Pentagonal Cylinders and Möbius Chains

Md. Abdus Sahir, Sk. Md. Abu Nayeem

Abstract

The degree-Kirchhoff index of a connected graph is defined as the sum of the reciprocals of the non-zero eigenvalues of the normalized Laplacian matrix, each multiplied by the graph's total degree. Several studies have recently obtained explicit formulations for the degree-Kirchhoff index of various kinds of class graphs. This paper presents closed-form formulas for the degree-Kirchhoff index of pentagonal cylinders and Möbius chains. Additionally, we calculate the Gutman index and Schultz index for these graphs.

Characterizing the Degree-Kirchhoff, Gutman, and Schultz Indices in Pentagonal Cylinders and Möbius Chains

Abstract

The degree-Kirchhoff index of a connected graph is defined as the sum of the reciprocals of the non-zero eigenvalues of the normalized Laplacian matrix, each multiplied by the graph's total degree. Several studies have recently obtained explicit formulations for the degree-Kirchhoff index of various kinds of class graphs. This paper presents closed-form formulas for the degree-Kirchhoff index of pentagonal cylinders and Möbius chains. Additionally, we calculate the Gutman index and Schultz index for these graphs.
Paper Structure (5 sections, 26 theorems, 76 equations, 3 figures, 1 table)

This paper contains 5 sections, 26 theorems, 76 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The derivation of degree-Kirchhoff index
  4. The relations among different indices
  5. Conclusion

Key Result

Theorem 1

Gut96Zhu96 Let $G$ be a simple and connected graph. The Kirchhoff index of $G$ is

Figures (3)

  • Figure 1: The pentagonal cylinder chain $P_n.$
  • Figure 2: The pentagonal Möbius chain $P'_n.$
  • Figure 3: Another representation of Möbius pentagonal chain $P'_n.$

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • Lemma 2: Matrix-determinant lemma
  • Lemma 3: Schur determinant formula
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • ...and 16 more