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

Abstract

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