Several illustrations of the Clar's aromatic sextet theory for the electronic properties of benzenoid hydrocarbons are demonstrated. It is shown how various techniques and concepts of the graph theory are useful for realizing and formulating not only this purely empirical theory but also the mathematical beauty of the structural formula of aromatic hydrocarbons. It is proposed that the sextet polynomial B _G(x) be defined in terms of the resonant sextet number p(G, k). For a thin polyhex graph, B _G(x) is shown to be equal to the number of the Kekulé structures K(G), while for fat polyhex the concept of the super-sextet needs to be introduced. Proper and improper sextets, Clar transformation, and sextet rotation are defined so that the relevant graph-theoretical manipulations can be transformed into algebra. By using the sextet polynomial and other graph-theoretical concepts thus defined, novel mathematical relations among several resonance-theoretical quantities which have been proposed by other researchers were found. Correlation between Clar's aromatic sextet and the benzene character proposed by Polansky and also the partial electron density map is pointed out.