On the spectrum of the normalized Laplacian of iterated triangulations of graphs

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.

Section: Introductionmentioning

confidence: 99%

On the normalized Laplacian of Möbius phenylene chain and its applications

Lei

Geng

et al. 2019

“…In this article, inspired by the works in Refs. [7, 23, 24, 26, and 28–31], we obtain the decomposition theorems for the Laplacian polynomial and normalized Laplacian polynomial of H n . Based on these results, explicit formulas for the Kirchhoff index, multiplicative degree‐Kirchhoff index and the number of spanning trees of H n are determined, respectively.…”

Section: Introductionmentioning

confidence: 99%

Kirchhoff index, multiplicative degree‐Kirchhoff index and spanning trees of the linear crossed hexagonal chains

Pan

2018

Let H n be a linear crossed hexagonal chain with n crossed hexagonals. In this article, we find that the Laplacian (resp. normalized Laplacian) spectrum of H n consists of the eigenvalues of a symmetric tridiagonal matrix of order 2n + 1 and a diagonal matrix of order 2n + 1. Based on the properties of these matrices, significant closed formulas for the Kirchhoff index, multiplicative degree-Kirchhoff index and the number of spanning trees of H n are derived. Finally, we show that the Kirchhoff (resp. multiplicative degree-Kirchhoff ) index of H n is approximately one quarter of its Wiener (resp. Gutman) index.

“…Recently, some closed‐form formulas for the multiplicative degree‐Kirchhoff index have been obtained for some types of graphs. One may be referred to References for detailed information.…”

Section: Introductionmentioning

confidence: 99%

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

Wei

Shi-qun

2018

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.