Frames generated by graphs

2024/05/27 by Deepshikha Deepshikha, Deepshikha · 3 citations
Computer Science · #05C50 #42C15 #42C40 #46C05 #Advanced Graph Theory Research #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2405.16891

openalex publication_date 2024/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Frames are the most natural generalization of orthonormal bases that allow the inclusion of redundant systems. In this article, we introduce the concept of frames generated by graphs in finite-dimensional spaces and study their properties. Let G be a simple graph of n vertices with Laplacian matrix L. We define the notions of G(n,k)-frames and L_G(n,k)-frames associated with the graph G. We obtain the family of dual frames of L_G(n,k)-frames and G(n,k)-frames. It is shown that non-regular graphs cannot generate tight frames. Then we establish a characterization of tight G(n,k)-frames in terms of the adjacency spectra of regular graphs. Besides, we provide a frame theoretic proof of an existing graph property. Finally, we show that one can use complete graphs to generate tight frames.

Cited by

• Optimal K dual frames and pairs in the presence of erasures
• Optimal Dual Frame Pairs: A Synergy with Graph Theory
• On spectrally optimal duals for r-erasures of frames generated by graphs

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