  We define the space of functions of bounded variation ($BV$) on the graph.
Using the notion of divergence of flows on graphs, we show that the unit ball
of the dual space to $BV$ in the graph setting can be described as the image of
the unit ball of the space $\ell^{\infty}$ by the divergence operator. Based on
this result, we propose a new iterative algorithm to find the exact minimizer
for the total variation (TV) denoising problem on the graph. The proposed
algorithm is provable convergent and its performance on image denoising
examples is compared with the Split Bregman and Primal-Dual algorithms as
benchmarks for iterative methods and with BM3D as a benchmark for other
state-of-the-art denoising methods. The experimental results show highly
competitive empirical convergence rate and visual quality for the proposed
algorithm.
