As the era of exascale supercomputing is coming, it is vital for next-generation supercomputers to find appropriate applications with high social and economic benefit. In recent years, it has been widely accepted that extremely-large graph computation is a promising killer application for supercomputing. Although Tianhe series supercomputers are leading in the world-wide competition of supercomputing (ranked No. 1 in the Top500 list for six times), previously they had been inefficient in graph computation according to the Graph500 list. This is mainly because the previous graph processing system cannot leverage the advanced hardware features of Tianhe supercomputers. To address the problem, in this paper we present our integrated optimizations for improving the graph computation performance on our next-generation exascale Tianhe supercomputing system, mainly including sorting with buffering for heavy vertices, vectorized searching with SVE (Scalable Vector Extension) on matrix2000+ CPUs, and group-based monitor communication on the proprietary interconnection network. Performance evaluation on a subset of the Tianhe exascale supercomputer (with 512 nodes and 96608 cores) shows that our customized graph processing system achieves 2131.98 GTEPS, which even outperforms the Tianhe-2 supercomputer (ranked No. 7 in Graph500 by running the state-of-the-art graph processing system) that has 16x more computing nodes.