4 research outputs found
Constructions of biangular tight frames and their relationships with equiangular tight frames
We study several interesting examples of Biangular Tight Frames (BTFs) -
basis-like sets of unit vectors admitting exactly two distinct frame angles
(ie, pairwise absolute inner products) - and examine their relationships with
Equiangular Tight Frames (ETFs) - basis-like systems which admit exactly one
frame angle.
We demonstrate a smooth parametrization BTFs, where the corresponding frame
angles transform smoothly with the parameter, which "passes through" an ETF
answers two questions regarding the rigidity of BTFs. We also develop a general
framework of so-called harmonic BTFs and Steiner BTFs - which includes the
equiangular cases, surprisingly, the development of this framework leads to a
connection with the famous open problem(s) regarding the existence of Mersenne
and Fermat primes. Finally, we construct a (chordally) biangular tight set of
subspaces (ie, a tight fusion frame) which "Pl\"ucker embeds" into an ETF.Comment: 19 page
A Characterization of Signed Graphs with Generalized Perfect Elimination Orderings
An important property of chordal graphs is that these graphs are
characterized by existence of perfect elimination orderings on their vertex
sets. In this paper, we generalize the notion of perfect elimination orderings
to signed graphs, and give a characterization for graphs admitting such
orderings, together with characterizations restricted to some subclasses and
further properties of those graphs.Comment: 18 pages; (v2) Reference updated (v3) Major update including title
change, shortening of proof of main theorem, addition of applications of main
theorem to special cases, reference updat