Clemente Gian Paolo, Cornaro Alessandra
Department of Mathematics and Econometrics, Catholic University, Milan, Italy.
J Inequal Appl. 2016;2016(1):285. doi: 10.1186/s13660-016-1234-6. Epub 2016 Nov 17.
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 in (Ars Math. Contemp. 5:99-115, 2012) proposed the notion of -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 -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.
在数学化学中，分子图邻接矩阵的中位数特征值与轨道能量和分子轨道密切相关。在这方面，已对最高能量占据轨道（HOMO）和最低能量未占据轨道（LUMO）之间的差异进行了研究（见Fowler和Pisansky所著《Acta Chim. Slov.》57:513 - 517, 2010）。受HOMO - LUMO分离问题的启发，Jaklič在（《Ars Math. Contemp.》5:99 - 115, 2012）中提出了 - 指数的概念，该指数用于衡量邻接矩阵中位数特征值的绝对值有多大。文献中已给出了该指数的若干界。本文的目的是推导用于界定 - 指数的其他不等式。通过应用优超技术并利用一些已知关系，我们为该指数推导了新的、更精确的上界。分析和数值结果展示了这些界在不同类图上的性能。
