The normalized Laplacian, degree-Kirchhoff index and the spanning tree numbers of generalized phenylenes

Recently, Peng and Li (2017) derived an explicit closed formula of Kirchhoff index and the number of spanning trees of linear phenylenes and their dicyclobutadieno derivatives, respectively in terms of the Laplacian spectrum. At the same time, they pointed that it is natural and interesting to study the normalized Laplacian, degree-Kirchhoff index and counting the spanning tree numbers of linear phenylenes and their dicyclobutadieno derivatives, respectively. Motivated by these, in this paper, explicit closed-form formulas for degree-Kirchhoff index and the number of spanning trees of generalized phenylenes are obtained based on the normalized Laplacian spectrum, respectively. (C) 2018 Elsevier B.V. All rights reserved.