The divergence in the computer architecture landscape has resulted in different architectures being considered mainstream at the same time. For application and algorithm developers, a dilemma arises when one must focus on using underlying architectural features to extract the best performance on each of these architectures, while writing portable code at the same time. We focus on this problem with graph analytics as our target application domain. In this paper, we present an abstraction-based methodology for performance-portable graph algorithm design on manycore architectures. We demonstrate our approach by systematically optimizing algorithms for the problems of breadth-first search, color propagation, and strongly connected components. We use Kokkos, a manycore library and programming model, for prototyping our algorithms. Our portable implementation of the strongly connected components algorithm on the NVIDIA Tesla K40M is up to 3.25x faster than a state-of-the-art parallel CPU implementation on a dual-socket Sandy Bridge compute node.