Frame graph

Tight Frame Graphs Arising as Line Graphs

“…It remains an open question (see [2]) whether the implication holds for graphs Γ with msr(Γ) > 2. In particular, since |L (K n )| = 1 2 n(n − 1), it remains to be determined whether L (K n ) is a tight frame graph for H d for n ≤ d ≤ ⌊ 1 4 n(n − 1)⌋. Although it is well known that the complete graph on n vertices is a tight frame graph in all possible dimensions, we provide an algorithmic proof here that utilizes a slight variant of the Laplacian method.…”

“…In this paper, we continue the line of investigation of [1], applying frame theoretic tools to a graph theoretic question. In the next section, we provide definitions and preliminary material from both graph theory and frame theory.…”

“…In [1] this problem was reintroduced from the point of view of frame theory. Frames in finitedimensional spaces have received much attention from both pure and applied mathematicians and constitute a vast literature; see [11] for a thorough introduction or [14] for a development at the undergraduate level.…”

“…and the Gramian operator Proof. As investigated in [1], a frame F = {f i } n i=1 for H d and a graph Γ = (V, E) with |V | = n may be associated with each other provided there is a one-to-one correspondence between vectors f i and vertices v i such that {v i , v j } ∈ E if and only if f i , f j = 0. This is known as a vector representation of the graph by a spanning set of H d .…”

A Characterization of Projective Unitary Equivalence of Finite Frames and Applications

“…We define the frame graph (cf. [4]) of a sequence of vectors (v j ) to be the graph with vertices {v j } (or the indices j themselves) and an edge between v j and v k , j = k ⇐⇒ v j , v k = 0.…”