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On the normalized Laplacian of Möbius phenylene chain and its applications
Author(s) -
Lei Lan,
Geng Xianya,
Li Shuchao,
Peng Yingjun,
Yu Yuantian
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.26044
Subject(s) - mathematics , multiplicative function , laplace operator , eigenvalues and eigenvectors , characteristic polynomial , degree (music) , quotient , laplacian matrix , combinatorics , phenylene , graph , pure mathematics , chain (unit) , order (exchange) , polynomial , mathematical analysis , physics , quantum mechanics , nuclear magnetic resonance , acoustics , polymer , finance , economics
Let L n 6 , 4denote a molecular graph of linear [ n ] phenylene with n hexagons and n squares, and let the Möbius phenylene chain HM n 6 , 4be the graph obtained from the L n 6 , 4by 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 HM n 6 , 4 , which consists of the eigenvalues of two symmetric matrices ℒ R and ℒ Q of order 3 n . 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 HM n 6 , 4 . Furthermore, we determine the limited value for the quotient of the multiplicative degree‐Kirchhoff index and the Gutman index of HM n 6 , 4 .