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

TL;DR

This paper characterizes walk-regular graphs and demonstrates that for frames generated by such graphs, the canonical dual frame is uniquely optimal for minimizing reconstruction error after erasures.

Contribution

It establishes the conditions under which the canonical dual frame is the unique optimal dual, linking graph properties to frame duality and erasure resilience.

Findings

Diagonal entries of the Moore-Penrose inverse are equal for walk-regular graphs.

Connected graphs generate full spark frames.

Canonical dual frames are uniquely optimal duals for frames from walk-regular graphs.

Abstract

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.