Characterization of some classes of graphs and their products as frame graph

TL;DR

This paper explores the relationship between the order of frame graphs and the dimension of the underlying space, focusing on specific classes of graphs and their products in the context of finite frames.

Contribution

It investigates how the order of frame graphs relates to the dimension of the space for various classes of graphs and their products.

Findings

Established connections between frame graph order and space dimension.

Analyzed specific classes of graphs and their products in the context of frames.

Provided insights into the structure of frame graphs in finite-dimensional spaces.

Abstract

To each finite frame $\varphi$ in an inner product space $\mathcal{H}$ we associate a simple graph $G(\varphi)$, called {\it frame graph}, with the vectors of the frame as vertices and there is an edge between vertices $f$ and $g$ provided that $\langle f, g\rangle \neq 0$ [1]. In this paper the relation between the order of $G(\varphi)$ and the dimension of $\mathcal{H}$ is investigated for some well-known classes of graphs and their products.