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 Ln be obtained from the transformation of the graph (Formula presented.) 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 Ln, then obtain its (multiplicative degree) Kirchhoff index and complexity corresponding to Ln. Finally, by comparing with its Wiener index and Gutman index, we are pleased to find that the (multiplicative degree) Kirchhoff index of Ln is nearly one quarter of its (Gutman) Wiener index.