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

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