Frames generated by graphs

TL;DR

This paper introduces the concept of frames generated by graphs in finite-dimensional spaces, characterizes their properties, and explores their relation to graph spectra and regularity, including conditions for tight frames.

Contribution

It defines new classes of graph-based frames, characterizes tight frames via graph spectra, and demonstrates how complete graphs can generate tight frames.

Findings

Non-regular graphs cannot generate tight frames

Tight G(n,k)-frames are characterized by the adjacency spectra of regular graphs

Complete graphs can be used to generate tight frames

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.