Study on the normalized Laplacian of a penta-graphene with applications

Let L-n denote a linear pentagonal chain with 2n pentagons. The penta-graphene (penta-C), denoted by R-n is the graph obtained from L-n by identifying the opposite lateral edges in an ordered way, whereas the pentagonal Mobius ring Rn ' is the graph obtained from the L-n by identifying the opposite lateral edges in a reversed way. In this paper, through the decomposition theorem of the normalized Laplacian characteristic polynomial and the relationship between its roots and the coefficients, an explicit closed-form formula of the multiplicative degree-Kirchhoff index (resp. Kemeny's constant, the number of spanning trees) of R-n is obtained. Furthermore, it is interesting to see that the multiplicative degree-Kirchhoff index of R-n is approximately 13 of its Gutman index. Based on our obtained results, all the corresponding results are obtained for R-n '.