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

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.