ON THE CONSTRUCTION AND PROPERTIES OF FRAMES USING INCIDENCE MATRIX OF GRAPHS AND THEIR SPECTRA

Abstract

Frames are considered to be redundant counterparts of bases for vector spaces. This redundant structure favours frames to be rich in both theory and applications. In recent studies on frames, graph theory is one of the significant tools to analyze the properties of different types of frames. In graph theory, we associate a graph with different types of matrices, of which signless Laplacian matrix contributes significantly in exploring the properties of a graph. In this paper, given a graph G, we propose a method to construct a frame from its incidence matrix such that its frame graph is the line graph of a derived graph of G. We analyze various properties of the frame constructed as above, its dual, etc. Further, we investigate the existence of frames with constrained frame bounds, using the properties of the associated graph and its signless Laplacian spectrum. © 2024 Jangjeon Research Institute for Mathematical Sciences and Physics. All rights reserved.