On the Multiplicative Degree-Kirchhoff Indices and the Number of Spanning Trees of Linear Phenylene Chains

Abstract

Let $L_n^{6,4,4}$ denote a molecular graph of linear $[n]$ phenylene, containing n hexagons and $2n-1$ squares. In this paper, using the decomposition theorem of the normalized Laplacian characteristic polynomial, we obtain that the normalized Laplacian spectrum of $L_n^{6,4,4}$ consisting of the eigenvalues of two symmetric tridiagonal matrices of order 4n is determined. By applying the relationship between the roots and coefficients of the characteristic polynomial of the above two matrices, an explicit closed-form formula of the multiplicative degree-Kirchhoff index (resp. the number of spanning trees) of $L_n^{6,4,4}$ is derived.

Keywords:

• Normalized Laplacian spectrum
• linear Phenylenes chain
• multiplicative degree-Kirchhoff index
• Spanning tree

AMS SUBJECT CLASSIFICATION:

• 05C50

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research is partially supported by National Science Foundation of China (Grant No. 11671164) and Natural Science Foundation of Anhui Province(Grant No. 2008085MA01).