The Quantitative Coronary Angiography (QCA) is a methodology used to evaluate the arterial diseases and, in particular, the degree of stenosis. In this paper we propose AQCA, a fully automatic method for vessel segmentation based on graph cut theory. Vesselness, geodesic paths and a new multi-scale edgeness map are used to compute a globally optimal artery segmentation. We evaluate the method performance in a rigorous numerical way on two datasets. The method can detect an artery with precision 92.9 +/- 5% and sensitivity 94.2 +/- 6%. The average absolute distance error between detected and ground truth centerline is 1.13 +/- 0.11 pixels (about 0.27 +/- 0.025 mm) and the absolute relative error in the vessel caliber estimation is 2.93% with almost no bias. Moreover, the method can discriminate between arteries and catheter with an accuracy of 96.4%.