2024/05/28 by Deepshikha Deepshikha, Deepshikha, Aniruddha Samanta +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · #05C40 #05C50 #42C15 #42C40 #46C05 #Combinatorics (math.CO) #FOS: Mathematics #Functional Analysis (math.FA) #Protein Tyrosine Phosphatases
paper · pdf · doi:10.48550/arxiv.2405.18189
openalex publication_date 2024/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.