The von Neumann graph entropy (VNGE) can be used as a measure of graph complexity, which can be the measure of information divergence and distance between graphs. However, computing VNGE is extensively demanding for a large-scale graph. We propose novel quadratic approximations for fast computing VNGE. Various inequalities for error between the quadratic approximations and the exact VNGE are found. Our methods reduce the cubic complexity of VNGE to linear complexity. Computational simulations on random graph models and various real network datasets demonstrate superior performance.