The normalized Laplacian, degree-Kirchhoff index and spanning trees of the linear polyomino chains

Let Bn be a linear polyomino chain with n squares. In this paper, according to the decomposition theorem of normalized Laplacian polynomial, we obtain that the normalized Laplacian spectrum of Bn   consists of the eigenvalues of two symmetric tridiagonal matrices of order n+1n+1. Together with the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, explicit closed formulas of the degree-Kirchhoff index and the number of spanning trees of Bn are derived. Furthermore, it is interesting to find that the degree-Kirchhoff index of Bn is approximately one half of its Gutman index.