On the degree-Kirchhoff index, Gutman index and the Schultz index of pentagonal cylinder/ M\"{o}bius chain

The degree-Kirchhoff index of a graph is given by the sum of inverses of non-zero eigenvalues of the normalized Laplacian matrix of the graph multiplied with the total degree of the graph. Explicit formulas for the degree-Kirchhoff index of various types of cylinder chain and M\"{o}bius chain have been obtained by many researchers in the recent past. In the present paper, we obtain closed-form formulas for the degree-Kirchhoff index of pentagonal cylinder/ M\"{o}bius chain. Also we find here the Gutman index and the Schultz index for those graphs