Calculating the normalized Laplacian spectrum and the number of spanning trees of linear pentagonal chains

Let Wn be a linear pentagonal chain with 2n pentagons. In this article, according to the decomposition theorem for the normalized Laplacian polynomial of Wn, we obtain that the normalized Laplacian spectrum of Wn consists of the eigenvalues of two special matrices: LA of order 3n+1 andLS of order 2n+1. Together with the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, explicit closed-form formulas for the degree-Kirchhoff index and the total number of spanning trees of Wn are derived, respectively. Finally, it is interesting to find that the degree-Kirchhoff index of Wn is approximately one half of its Gutman index.