On the Laplacians and Normalized Laplacians for Graph Transformation with Respect to the Dicyclobutadieno Derivative of [n]Phenylenes

Abstract

As a crucial tool for exploring and characterizing the structure properties of the molecular graphs, spectrum analysis and computation are flourishing in recent year. Let L_n be obtained from the transformation of the graph $L_n^{6,4},$ which obtained by attaching crossed four-membered rings to the terminal of crossed phenylenes. In this article, we first determine the (normalized) Laplacian spectra of L_n, then obtain its (multiplicative degree) Kirchhoff index and complexity corresponding to L_n. Finally, by comparing with its Wiener index and Gutman index, we are pleased to find that the (multiplicative degree) Kirchhoff index of L_n is nearly one quarter of its (Gutman) Wiener index.

Graphical Abstract

Keywords:

• Complexity
• dicyclobutadieno derivative
• Gutman index
• (multiplicative degree) Kirchhoff index
• Wiener index

Disclosure statement

There are no conflicts of interest.

Additional information

Funding

This work was supported in part by Anhui Provincial Natural Science Foundation under Grant 2008085J01, and by National Natural Science Foundation of China Grant 11601006, and by China Postdoctoral Science Foundation under Grant 2017M621579.