Sparse matrix-vector multiplication (SpMV) is a critical building block for iterative graph analytics algorithms. Typically, such algorithms have a varying active vertex set across iterations. This variability has been used to improve performance by either dynamically switching algorithms between iterations (software) or designing custom accelerators (hardware) for graph analytics algorithms. In this work, we propose a novel framework, CoSPARSE, that employs hardware and software reconfiguration as a synergistic solution to accelerate SpMV-based graph analytics algorithms. Building on previously proposed general-purpose reconfigurable hardware, we implement CoSPARSE as a software layer, abstracting the hardware as a specialized SpMV accelerator. CoSPARSE dynamically selects software and hardware configurations for each iteration and achieves a maximum speedup of 2.0$\times$ compared to the na"ive implementation with no reconfiguration. Across a suite of graph algorithms, CoSPARSE outperforms a state-of-the-art shared memory framework, Ligra, on a Xeon CPU with up to 3.51$\times$ better performance and 877$\times$ better energy efficiency.