Du, Ni; Li, Xuechao The normalized Laplacian, degree-Kirchhoff index and spanning trees of the linear ladder-like chains. (English) Zbl 1427.05132 Adv. Appl. Discrete Math. 20, No. 1, 133-164 (2019). Summary: This paper follows up the problem posted in [J. Huang et al., Discrete Appl. Math. 207, 67–79 (2016; Zbl 1337.05058)]. Let \(A_{2n}\) be a linear Ladder-like polyomino chain of \(2n\) vertices with \(n-4\) squares. In this paper, according to the decomposition theorem of normalized Laplacian polynomial, we obtain the normalized Laplacian spectrum of \(A_{2n}\) consists of the eigenvalues of two symmetric tridiagonal matrices of order n. Together with the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, explicit formulas for one case of the degree-Kirchhoff index and the number of spanning trees of \(A_{2n}\) are derived. MSC: 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.) 05C07 Vertex degrees 05C31 Graph polynomials Keywords:linear polyomino chain; normalized Laplacian; degree-Kirchhoff index; spanning tree Citations:Zbl 1337.05058 PDFBibTeX XMLCite \textit{N. Du} and \textit{X. Li}, Adv. Appl. Discrete Math. 20, No. 1, 133--164 (2019; Zbl 1427.05132) Full Text: DOI References: [1] E. Bendito, A. Carmona, A. M. Encinas and J. M. Gesto, A formula for the Kirchhoff index, Int. J. Quantum Chem. 108 (2008), 1200-1206. · Zbl 1224.39008 [2] M. Bianchi, A. Cornaro, J. L. Palacios and A. Torriero, Bounds for the Kirchhoff index via majorization techniques, J. Math. Chem. 51 (2013), 569-587. · Zbl 1327.05066 [3] S. B. Bozkurt and D. Bozkurt, On the sum of powers of normalized Laplacian eigenvalues of graphs, MATCH Commun. Math. Comput. Chem. 68 (2012), 917930. · Zbl 1289.05269 [4] A. Carmona, A. M. 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