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Two weighted inequalities for real non-negative sequences are proven. The first one represents a generalization of the Szőkefalvi Nagy inequality for the variance, and the second a generalization of the discrete Chebyshev inequality for two real sequences. Then, the obtained inequalities are used to determine lower bounds for some degree-based topological indices of graphs. [ABSTRACT FROM AUTHOR]

The McClelland formula, based on the upper bound $$\sqrt{2mn}$$ , is capable of reproducing over 99.5% of the total $$\pi $$ -electron energy ( $$E_\pi $$ ) of a conjugated hydrocarbon, whose molecules possess n carbon atoms and m carbon-carbon bonds. Its weak point is that it predicts equal $$E_\pi $$ -values for all isomers. We now show how this failure can be overcome, offering a general strategy for extending McClelland's formula. By means of one of these extensions, $$E_\pi $$ is related with the energy of the highest occupied molecular orbital, and the error of the new formula is diminished by more than $$50\%$$ relative to the standard McClelland approximation. [ABSTRACT FROM AUTHOR]

A relation is obtained between the total π -electron energy E π and the HOMO energy E HOMO , valid within the HMO approximation. This seems to be the very first relation between E π and E HOMO ever established. It enables a much more accurate assessment of E π of alternant conjugated hydrocarbons than that based on McClelland's formula and on other ( n , m )-type approximate expressions. [ABSTRACT FROM AUTHOR]