The Laplacians and Normalized Laplacians of the linear chain networks
and applications
Abstract
In recent years, spectrum analysis and computation have developed
rapidly in order to explore and characterize the properties of network
sciences. Let Ln be obtained from the transformation of the graph L6,4,4
n , which obtained by attaching crossed two four-membered rings to the
terminal of crossed phenylenes. Firstly, we study the (nomalized)
Laplacian spectrum of Ln based on the decomposition theorem for the
corresponding matrices. Secondly, we obtain the closed-term fomulas for
the (multiplicative degree) Kirchhoff index and the number of spanning
trees from the relationship between roots and coefficients in linear
chain networks. Finally, we are surprised to find that the
(multiplicative degree) Kirchhoff index of Ln is nearly to one quarter
of its (Gutman) Wiener index when n tends to infinity.