On walk-regular graphs and optimal duals of frames generated by graphs

Erasures are a common problem that arises while signals or data are being transmitted. A profound challenge in frame theory is to find the optimal dual frames ($OD$-frames) to minimize the reconstruction error if erasures occur. In this paper, we study the optimal duals of frames generated by graphs. First, we characterize walk-regular graphs. Then, it is shown that the diagonal entries of the Moore-Penrose inverse of the Laplacian matrix (or adjacency matrix) of a walk-regular graph are equal. Besides, we prove that connected graphs generate full spark frames. Using these results, we establish that the canonical dual frames are the unique $OD$-frames of a frame generated by a walk-regular graph. A sufficient condition under which the canonical dual frame is the unique $OD$-frame is known. Here, we establish that the condition is also necessary if the frame is generated by a connected graph.