Estimating the total π-electron energy

The paper gives a short survey of the most important lower and upper bounds for the total π-electron energy, i.e., the graph energy (E). In addition, a new lower and a new upper bound for E are deduced, valid for general molecular graphs. The strengthened versions of these estimates, valid for alternant conjugated hydrocarbons, are also reported. Copyright 2013 (CC) SCS

Positronium – Hydrogen Like and Unlike

On the occasion of the 2008 Brijuni Conference on Hydrogen – the most abundant atomic species in the Universe, it seems fitting to draw attention of the participants of this conference, as well as chemists at large, to Positronium – one of the least abundant atom-like species in the Universe, if for no other reasons then because it was theoretically predicted by a Croatian scientist, Stjepan Mohorovičić some 75 years ago, about 100 miles away, in the city of Zagreb, the capitol of the Republic of Croatia. Abstract. A brief review on positronium, Ps, hydrogen-like system built from positron and electron, is outlined from its beginning in 1935, the first theoretical study on this relatively stable matter-antimatter system by Stjepan Mohorovičić, to the most recent works on positronim hydride PsH, and positronium molecule Ps2, analogue of hydrogen molecule. Mohorovičić calculated spectra of Ps and was even looking for it in the sky searching for its spectrum, but experimental observations of positronium Lyman-α radiation Lyα λ2430 line waited for another 40 years before being successful identified in a laboratory in 1975 by Canter and collaborators. It took another ten years for astronomical observation of the presence of positronium in outer space in 1984 by McClintock, who observed Lyα λ2430 line in spectra of Crab Nebula, 50 years after the attempts of S. Mohorovičić to detect positronium lines. The work of Mohorovičić was mostly ignored in his native Croatia, until the most recent time, an illustration of “historical blunder” of local physics community – phenomenon not so unheard of in science in general, as has been recently worldwide illustrated with hesitation of acceptance of the notion of nonlinear dose response (hormesis); the density functional theory; and chemical graph theory.</p

Invariantes espetrais da matriz de Randic de um grafo

Este trabalho apresenta um estudo sobre invariantes espetrais da matriz de Randić de um grafo. O índice de Randić é um invariante espetral apresentado em 1975 por Milan Randić e com importantes aplicações ao nível da Química. Em 2010 define-se a matriz de Randić, uma matriz não negativa construída a partir desse índice. O estudo do espetro de matrizes associadas a grafos é um dos grandes objetivos da investigação em teoria dos grafos e são já diversas as aplicações em diferentes áreas científicas. Neste trabalho é estudado o espetro da matriz de Randić associada a um grafo e definido o spread de Randić. Para além disso, são apresentados majorantes e minorantes para esse invariante espetral. Em Química, a energia de grafos caterpillar, que estão associados a sistemas aromáticos, está relacionada com as relações de ressonância desses sistemas. Tendo esse facto como motivação, é estudado o espetro e o espetro de Randić de grafos caterpillar e são apresentados majorantes para a energia e para a energia de Randić dessa classe de grafos.This work presents a study related to spectral invariants for the Randić matrix of a graph. The Randić index is a spectral invariant presented in 1975 by Milan Randić and with important applications in chemistry. In 2010 the Randić matrix was defined as a nonnegative matrix built from this index. The study of the spectrum of matrices associated with graphs is one of the major goals of research in graph theory and there are already several applications in different scientific areas. In this work the spectrum of the Randić matrix associated to a graph is studied and the Randić spread is defined. In addition, upper and lower bounds are presented for this spectral invariant. In chemistry, the energy of caterpillar graphs, that are associated with aromatic systems, is related with the resonance of these systems. Having this as motivation, the spectrum and the Randić spectrum of caterpillar graphs are studied and upper bounds are presented for the energy and for the Randić energy of this class of graphs.Programa Doutoral em Matemátic