Abstract
The normalized Laplacian is extremely important for analyzing the structural properties of non-regular graphs. The molecular graph of generalized phenylene consists of n hexagons and 2n squares, denoted by Ln6,4,4. In this paper, by using the normalized Laplacian polynomial decomposition theorem, we have investigated the normalized Laplacian spectrum of Ln6,4,4 consisting of the eigenvalues of symmetric tri-diagonal matrices LA and LS of order 4n+1. As an application, the significant formula is obtained to calculate the multiplicative degree-Kirchhoff index and the number of spanning trees of generalized phenylene network based on the relationships between the coefficients and roots.
Highlights
Introduction and PreliminariesKirchhoff Index and Spanning Tree ofIn this paper, only simple, undirected, and finite graphs are considered
In 2007, Chen and Zhang [8] presented a new index based on normalized Laplacian, which is called the multiplicative degree-Kirchhoff index and defined it as, K f ∗ ( G ) = 2| EG | ∑in=2 λ1
We resolve to discover the calculation for multiplicative degree-Kirchhoff index of Ln
Summary
Only simple, undirected, and finite graphs are considered. Assume that. In [2], a well-known distance based parameter named as Wiener index and denoted by WG is introduced for the first time. This parameter is obtained by adding the distances between every pair of vertices in G, that is, WG = ∑{vi ,v j }⊆VG dij. In 2007, Chen and Zhang [8] presented a new index based on normalized Laplacian, which is called the multiplicative degree-Kirchhoff index and defined it as, K f ∗ ( G ) = 2| EG | ∑in=2 λ1. By using the decomposition theorem for the normalized Laplacian characteristic polynomial, the multiplicative degree-Kirchhoff index, as well as spanning trees of L6,4,4 are given explicit closed-form formulas
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