Close-to-zero eigenvalues of the rooted product of graphs
In the quantum chemistry the energy gap refers to the energy of an electron in the HOMO-LUMO transfer. The energy gap is the difference between the smallest positive \(\lambda_+\) and the largest negative eigenvalue \(\lambda_-\) of the adjacency matrix of the associated molecular graph. In this paper the authors use a rooted product of two graphs \(G\) and \(H\) denoted \(G(H)\) and they name it a vertex-decorated graph. The main idea is to choose graph \(H\) for a given graph \(G\) in such a way that the close to zero eigenvalues \(\lambda_+\) and \(\lambda_-\) of the rooted product \(G(H)\) give the smallest difference. For a certain family of graphs the characterization of obtaining a close to zero energy gap is given. The method is presented on four examples.
scientific article; zbMATH DE number 7369403