Characterizing the Degree-Kirchhoff, Gutman, and Schultz Indices in Pentagonal Cylinders and M\"{o}bius Chains

TL;DR

This paper derives explicit formulas for the degree-Kirchhoff, Gutman, and Schultz indices of pentagonal cylinders and Möbius chains, advancing the understanding of these graph invariants in specific molecular structures.

Contribution

It provides the first closed-form formulas for these indices in pentagonal cylinders and Möbius chains, expanding the analytical tools for these graph classes.

Findings

Closed-form formulas for degree-Kirchhoff index

Explicit calculations of Gutman and Schultz indices

Enhanced understanding of graph invariants in molecular structures

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 M\"{o}bius chains. Additionally, we calculate the Gutman index and Schultz index for these graphs.