On the normalized Laplacian of Möbius phenylene chain and its applications

Abstract

Let Ln6,4 denote a molecular graph of linear [n] phenylene with n hexagons and n squares, and let the Möbius phenylene chain H0emMn6,4 be the graph obtained from the Ln6,4 by identifying the opposite lateral edges in reversed way. Utilizing the decomposition theorem of the normalized Laplacian characteristic polynomial, we study the normalized Laplacian spectrum of H0emMn6,4, which consists of the eigenvalues of two symmetric matrices ℒ R and ℒ Q of order 3n. By investigating the relationship between the roots and coefficients of the characteristic polynomials of the two matrices above, we obtain an explicit closed‐form formula of the multiplicative degree‐Kirchhoff index as well as the number of spanning trees of H0emMn6,4. Furthermore, we determine the limited value for the quotient of the multiplicative degree‐Kirchhoff index and the Gutman index of H0emMn6,4.

The Laplacians and Normalized Laplacians of the linear chain networks and applications

“…Ma [30] got the normalized Laplacian spectrum of linear phenylene, and the linear phenylene containing has n hexagons and n − 1 squares. L. Lan [31] explored the linear phenylene with n hexagons and n squares. Umar Ali [32] analyzed the strong prism of a graph G is the strong product of the complete graph of order 2 and G. X.Y.…”

Kirchhoff index and complexity of linear Möbius and cylinder octagonal‐quadrilateral networks with respect to Laplacians

“…In 2019, Geng et al [14] used this method to calculate the Kirchhoff index of the derived Möbius chain and cylinder chain of a phenylenes chain. Lei et al [15] determined the degree-Kirchhoff index of Möbius phenylene chain. Shi and Liu et al [16] computed the Kirchhoff index of linear octagonal-quadrilateral network in 2020.…”

Kemeny's constant and Kirchhoffian indices for a family of non-regular graphs

“…The study of Kirchhoff indices and related quantities to do with electric resistances on graphs is an active research field. Recent works related to the results of this paper include [1,4,5,7,9,14,15,24,25].…”