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