title: Bounding theHL-index of a graph: a majorization approach.
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, Jakličet al.in (Ars Math. Contemp. 5:99-115, 2012) proposed the notion ofHL-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 theHL-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.
authors.affiliation: Department of Mathematics and Econometrics, Catholic University, Milan, Italy.
authors.forename: Gian Paolo
authors.initials: GP
authors.lastname: Clemente
authors.affiliation: Department of Mathematics and Econometrics, Catholic University, Milan, Italy.
authors.forename: Alessandra
authors.identifier: 0000-0002-8137-9440
authors.initials: A
authors.lastname: Cornaro
keywords: HOMO-LUMO; graph eigenvalue; majorization; median eigenvalue