We have investigated the correlations between the equivalent widths of 21 selected diffuse interstellar bands (DIBs) and the corresponding interstellar column densities N (C 2), N (CN), and N (CH), toward 53 stars with color excesses 0.11≤ E (B-V)≤ 1.99. The observational data were derived primarily from echelle spectra acquired at R= 38,000 as part of our extensive, continuing survey of the bands. All but six of the 53 final spectra show signal-to-noise ratios≥ 800 at 5780 Å. The principal result presented here is that seven of the 21 bands prove to be examples of" the C 2 DIBs," a class of weak, narrow bands whose normalized equivalent widths W λ (X)/W λ (λ 6196) are well correlated specifically with N (C 2)/E (B-V) via power laws. In contrast, the similarly normalized equivalent widths of the 14 other, well-known DIBs analyzed here are uncorrelated, or weakly anticorrelated, with N (C 2)/E (B-V), to within the observational uncertainties. Thus, the polyatomic molecule (s) presumed to cause these seven C 2 DIBs may bear a direct chemical relation to C 2 that is not shared by the polyatomic molecules putatively responsible for the other 14 bands. The C 2 DIBs also show positive correlations with N (CN)/E (B-V) and N (CH)/E (B-V) in our particular sample of light paths, although generally with shallower slopes in the case of N (CN) and with greater scatter in the case of N (CH). Eleven additional C 2 DIBs are also identified but are not analyzed here. Among the 18 C 2 DIBs identified, four apparently have not been previously detected. The λ 4963 band is generally the strongest of the 18 C 2 DIBs, while the λ 4734 band shows the most sensitive correlation with N (C 2).