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This text is directed toward senior undergraduate and beginning graduate students as well as practicing chemists. It provides conceptual tools for understanding chemical phenomena with HMO theory. The author demonstrates that one can use graph theory to streamline HMO calculations without the aid of group theory or computer programmes. The merging of graph theory and chemical theory is intuitive to chemists. Chemical graph theory provides simple descriptive interpretations of complicated quantum mechanical calculations. The author provides practical applications on numerous HMO energy levels (eigenvalues) and corresponding wave functions (eigenvectors).