Laplacians and equioverlapping vectors

Igor Balla, Filip Kučerák, Xichao Shu

Session: Optimal frames and codes (Dustin Mixon, Matthew Fickus)

Keywords: eigenvalues, algebraic connectivity, Laplacian, equioverlapping, semidefinite

Abstract: Real equiangular lines or equivalently, spherical $\{\pm \alpha\}$-codes, and their correspondence to adjacency matrices of graphs have been studied extensively. In this talk, we will discuss a generalization of this correspondence where the adjacency matrix is replaced by the Laplacian matrix of a graph. The spherical $\{\pm \alpha\}$-codes are, in turn, replaced by more general objects which we refer to as equioverlapping vectors, in analogy to the notion of equioverlapping measurements studied in quantum information theory. We will discuss how this correspondence can be used to obtain new bounds on the size and multiplicity of the first nonzero eigenvalue of a graph Laplacian.

Submission Number: 35