The normalized Laplacians, degree-Kirchhoff index and the spanning trees of linear hexagonal chains

Let LnLn be a linear hexagonal chain with nn hexagons. In this paper, according to the decomposition theorem of normalized Laplacian polynomial of a graph, we obtain that the normalized Laplacian spectrum of LnLn consists of the eigenvalues of two symmetric tridiagonal matrices of order 2n+12n+1. Together with the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, explicit closed formula of the degree-Kirchhoff index (resp. the number of spanning trees) of LnLn is derived. Finally, it is interesting to find that the degree-Kirchhoff index of LnLn is approximately one half of its Gutman index.