Premium
On normalized Laplacians, multiplicative degree‐Kirchhoff indices, and spanning trees of the linear [ n ]phenylenes and their dicyclobutadieno derivatives
Author(s) -
Li Shuchao,
Wei Wei,
Yu Shiqun
Publication year - 2019
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.25863
Subject(s) - mathematics , multiplicative function , degree (music) , combinatorics , spanning tree , eigenvalues and eigenvectors , resistance distance , laplace operator , laplacian matrix , graph , mathematical analysis , physics , line graph , quantum mechanics , acoustics , graph power
Abstract 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.