On walk-regular graphs and optimal duals of frames generated by graphs
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910461046292480 |
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| author | Deepshikha Samanta, Aniruddha |
| author_facet | Deepshikha Samanta, Aniruddha |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_18189 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On walk-regular graphs and optimal duals of frames generated by graphs Deepshikha Samanta, Aniruddha Combinatorics Functional Analysis 05C50, 05C40, 42C15, 42C40, 46C05 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. |
| title | On walk-regular graphs and optimal duals of frames generated by graphs |
| topic | Combinatorics Functional Analysis 05C50, 05C40, 42C15, 42C40, 46C05 |
| url | https://arxiv.org/abs/2405.18189 |