On Normalized Laplacian, Degree-Kirchhoff Index of the Strong Prism of Generalized Phenylenes^*

Abstract

The eigenvalues of normalized Laplacian are extremely important for analyzing the structure properties of non-regular graphs. A molecular graph of generalized phenylene L_n is a linear chain containing n hexagons and n quadrangles. The strong prism of a graph G is the strong product of the complete graph of order 2 and G. Let $L_n^2$ be the strong prism of generalized phenylene L_n. In this paper, the normalized Laplacian spectra of $L_n^2$ were scrutinized by using decomposition theorem for the corresponding matrices. Further, the explicit formulae of the degree-Kirchhoff index and the number of spanning trees for $L_n^2$ were obtained.

Keywords:

• Generalized phenylene
• normalized Laplacian
• multiplicative degree-Kirchhoff index
• spanning tree

MSC Subject Classification:

• 05C50
• 05C90

Acknowledgments

The authors would like to express their sincere gratitude to the anonymous referees for valuable suggestions, which led to a great deal of improvement in the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).