Abstract
In mathematical chemistry, the median eigenvalues of the adjacency matrix of a molecular graph are strictly related to orbital energies and molecular orbitals. In this regard, the difference between the occupied orbital of highest energy (HOMO) and the unoccupied orbital of lowest energy (LUMO) has been investigated (see Fowler and Pisansky in Acta Chim. Slov. 57:513-517, 2010). Motivated by the HOMO-LUMO separation problem, Jaklič et al. in (Ars Math. Contemp. 5:99-115, 2012) proposed the notion of HL-index that measures how large in absolute value are the median eigenvalues of the adjacency matrix. Several bounds for this index have been provided in the literature. The aim of the paper is to derive alternative inequalities to bound the HL-index. By applying majorization techniques and making use of some known relations, we derive new and sharper upper bounds for this index. Analytical and numerical results show the performance of these bounds on different classes of graphs.
Highlights
The Hückel molecular orbital method (HMO) is a methodology for the determination of energies of molecular orbitals of π -electrons
The contribution of this paper is along those lines: we derive, through a methodology based on majorization techniques, new bounds on the median eigenvalues of the normalized Laplacian matrix
5 Conclusions In this paper alternative upper bounds on the HL-index are provided by means of majorization techniques
Summary
The Hückel molecular orbital method (HMO) (see [ ]) is a methodology for the determination of energies of molecular orbitals of π -electrons. The contribution of this paper is along those lines: we derive, through a methodology based on majorization techniques (see [ – ] and [ ]), new bounds on the median eigenvalues of the normalized Laplacian matrix. We employ a theoretical methodology proposed by Bianchi and Torriero in [ ] based on nonlinear global optimization problems solved through majorization techniques These bounds can be quantified by using the numerical approaches developed in [ ] and [ ] and extended for the normalized Laplacian matrix in [ ] and in [ ]. We take advantage of an existing bound on the energy index (see [ ]) depending on additional information on the first eigenvalue of the adjacency matrix This additional information is obtained here by using majorization techniques in order to provide new inequalities for the HL-index of bipartite and nonbipartite graphs. Section shows how the bound determined in Section . improves those presented in the literature
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