Graph applications are employed in many fields but show poor performance on general-purpose computing systems due to heavy, irregular, and data-driven memory access patterns. The diverse topology of real-life graphs also affects the performance. Even though many hardware accelerators have been proposed to mitigate performance issues and provide energy efficiency, programmability and flexibility have not been sufficiently addressed. This thesis presents a domain-specific processor design based on extending the RISC-V Instruction Set Architecture (ISA). The proposed approach uses new instructions supported by the compiler and software library. Micro-architectural design for executing the new instructions is based on a scratchpad-memory (SPM), prefetcher, and a non-blocking cache system. The custom processor is implemented using System Verilog HDL and simulated with Xilinx's Vivado Design Suite. LLVM Compiler Framework is used for compiler support and optimization. The software library for utilizing the custom instructions uses Gather-Apply-Scatter (GAS) paradigm. The system is evaluated on well-known graph benchmarks, while sensitivity analysis is done on various parameters for achieving the best performance with minimal cost. Performance is measured on both native benchmarks and the software library. In addition, compiler support is evaluated for its effect on performance. Cost-efficient performance evaluations show average speedups between 10% and 49% for different benchmarks, while the single-core architecture can achieve up to 73%.