Double gaussianization of graph spectra

The graph spectrum is the set of eigenvalues of a simple graph with n vertices. Here we fold this graph spectrum at a given pair of reference eigenvalues and then exponentiate the resulting folded graph spectrum. This process produces double Gaussianized functions of the graph adjacency matrix which give more importance to the reference eigenvalues than to the rest of the spectrum. Based on evidences from mathematical chemistry we focus here our attention on the reference eigenvalues ?1 . In the examples that we have examined, they enclose most of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of organic molecular graphs. We prove here several results for the trace of the double Gaussianized adjacency matrix of simple graphs? the double Gaussianized Estrada index. Finally we apply this index to the classification of polycyclic aromatic hydrocarbons (PAHs) as carcinogenic or inactive ones. We discover that local indices based on the previously developed matrix function allow to classify correctly 100% of the PAHs analyzed. Such indices reflect the electron population of the HOMO/LUMO and eigenvalues close to them, in the so-called K and L regions of PAHs. ? 2020 Elsevier Inc. All rights reserved.

Automated summary

The study focuses on folding and exponentiating the graph spectrum, showing that the double Gaussianized functions highlight the importance of reference eigenvalues. By concentrating on the reference eigenvalues λ1 based on evidence from mathematical chemistry, it is discovered that they encompass the HOMO and LUMO. Finally, applying this method to the classification of polycyclic aromatic hydrocarbons resulted in a successful classification of all analyzed compounds.