 Solution

Each row of the oriented incidence matrix records one signed endpoint difference, henceReversing an edge changes the sign of its row, and permuting the edges permutes rows; neither operation changes the sum of absolute differences. Thus the graph total variation is independent of the chosen orientation and ordering. If $D\theta =0$, values agree at the endpoints of every edge. Any two graph vertices of a connected graph are joined by a graph path, so their values agree. Therefore $\ker D=\mathrm{span}\{1\}$.