We present experimental and numerical results for the fluctuation properties in the eigenfrequency spectra and of the scattering matrix of closed and open unidirectional quantum graphs, respectively. Unidirectional quantum graphs, that are composed of bonds connected by reflectionless vertices, were introduced by Akila and Gutkin [Akila and Gutkin, J. Phys. A: Math. Theor. 48, 345101 (2015)1751-811310.1088/1751-8113/48/34/345101]. The nearest-neighbor spacing distribution of their eigenvalues was shown to comply with random-matrix theory predictions for typical chaotic systems with completely violated time-reversal invariance. The occurrence of short periodic orbits confined to a fraction of the system, that lead in conventional quantum graphs to deviations of the long-range spectral correlations from the behavior expected for typical chaotic systems, is suppressed in unidirectional ones. Therefore, we pose the question whether such graphs may serve as a more appropriate model for closed and open chaotic systems with violated time-reversal invariance than conventional ones. We compare the fluctuation properties of their eigenvalues and scattering matrix elements and observe especially in the long-range correlations larger deviations from random-matrix theory predictions for the unidirectional graphs. These are attributed to a loss of complexity of the underlying dynamic, induced by the unidirectionality.