On normalized Laplacians, multiplicative degree-Kirchhoff indices, and spanning trees of the linear [n]phenylenes and their dicyclobutadieno derivatives

Let Ln6,6 be the molecular graph of the linear [n]phenylene with n hexagons and n - 1 squares, and let Ln4,4 be the graph obtained by attaching four-membered rings to the terminal hexagons of Ln6,6. In this article, the normalized Laplacian spectrum of Ln6,6 consisting of the eigenvalues of two symmetric tridiagonal matrices of order 3n is determined. An explicit closed-form formula of the multiplicative degree-Kirchhoff index (respectively the number of spanning trees) of Ln6,6 is derived. Similarly, explicit closed-form formulas of the multiplicative degree-Kirchhoff index and the number of spanning trees of Ln4,4 are obtained. It is interesting to see that the multiplicative degree-Kirchhoff index of Ln6,6 (respectively Ln4,4) is approximately to one half of its Gutman index.