We investigate asymptotic consensus of linear systems under a class of switching communication graphs. We significantly relax several reciprocity and connectivity assumptions prevalent in the consensus literature by employing switched-systems techniques to establish consensus. Our results rely solely on asymptotic properties of the switching communication graphs in contrast to classical average dwell-time conditions. A bound on the uniform rate of convergence to consensus is also established as part of this work. (C) 2018 Elsevier Ltd. All rights reserved.