Context. The identification of the carriers of the diffuse interstellar bands (DIBs) is the longest standing problem in the study of the interstellar medium. Here we present recent UV laboratory spectra of various polycyclic aromatic hydrocarbons (PAHs) and explore the potential of these molecules as carriers of the DIBs. Whereas, in the near IR range, the PAHs exhibit vibrational bands that are not molecule-specific, their electronic transitions occurring in the UV/vis provide characteristic fingerprints. The comparison of laboratory spectra calibrated in intensity with high signal-to-noise observational data in the UV enables us to establish new constraints on PAH abundances. Aims. From a detailed comparison of the gas-phase and Ne-matrix absorption spectra of anthracene, phenanthrene, pyrene, 2,3-benzofluorene, benzo[ghi]perylene, and hexabenzocoronene with new interstellar spectra, we aim to infer the abundance of these PAHs in the interstellar medium. Methods. We present spectra of PAHs measured at low temperature in the gas phase and in an Ne matrix, and present methods to derive absolute absorption cross sections for the matrix and gas phase spectra. We have obtained high signal to noise (S/N > 100) absorption spectra toward five lines of sight with reddenings of E B−V = 1−1.6 mag. The spectra cover the 3050−3850 Å wavelength region where the PAHs studied here show prominent absorption features. Results. From the observations, we infer upper limits in the fractional abundances of the PAHs studied here. Upper limits in the column densities of anthracene of 0.8−2.8 × 10 12 cm −2 and of pyrene and 2,3-benzofluorene ranging from 2−8 × 10 12 cm −2 are inferred. Upper limits in the column densities of benzo [ghi]perylene are 0.9−2.4 × 10 13 and 10 14 cm −2 for phenanthrene. The measurements indicate fractional abundances of anthracene, pyrene, and 2,3-benzofluorene of a few times 10 −10 . Upper limits in the fractional abundance of benzo [ghi]perylene of a few times 10 −9 and of phenanthrene of few times 10 −8 are inferred. Toward CPD −32 • 1734, we found near 3584 Å an absorption line of OH + , which was discovered in the interstellar medium only very recently. Conclusions. The fractional abundances of PAHs inferred here are up to two orders of magnitude lower than estimated total PAH abundances in the interstellar medium. This indicates that either neutral PAHs are not abundant in translucent molecular clouds or that a PAH population with a wide variety of molecules is present.