To promote possible applications of graphene in molecular identification based on stacking effects, in particular in recognizing aromatic amino acids and even sequencing nucleobases in life sciences, we comprehensively study the interaction between graphene segments and different cyclic organic hydrocarbons including benzene (C6H6), cyclohexane (C6H12), benzyne (C6H4), cyclohexene (C6H10), 1,3-cyclohexadiene (C6H8(1)) and 1,4-cyclohexadiene (C6H8(2)), using the density-functional tight-binding (DFTB) method. Interestingly, we find obviously different characteristics in Raman vibrational and ultraviolet visible absorption spectra of the small molecules adsorbed on the graphene sheet. Specifically, we find that both spectra involve clearly different characteristic peaks, belonging to the different small molecules upon adsorption, with the ones of ionized molecules being more substantial. Further analysis shows that the adsorptions are almost all due to the presence of dispersion energy in neutral cases and involve charge transfer from the graphene to the small molecules. In contrast, the main binding force in the ionic adsorption systems is the electronic interaction. The results present clear signatures that can be used to recognize different kinds of aromatic hydrocarbon rings on graphene sheets. We expect that our findings will be helpful for designing molecular recognition devices using graphene.