Speaker: Edwin van Dam (Tilburg University )
|Date||10 November 2021|
Unit gain graphs and lines in complex space with few angles
Since the introduction of the Hermitian adjacency matrix for digraphs by Guo and Mohar, interest in so-called complex unit gain graphs has surged. In this talk, we consider such gain graphs with two distinct eigenvalues. Analogously to (undirected) graphs whose traditional adjacency matrix has few distinct eigenvalues, a great deal of structural symmetry is required. Besides combinatorial considerations, also the representation by lines in complex space is essential in the study of considered gain graphs. Examples are drawn from various relevant concepts from quantum information theory related to lines in complex space with few angles, such as SIC-POVMs and MUBs.
Other examples relate to the hexacode, Coxeter-Todd lattice, and the Van Lint-Schrijver association scheme. Many other examples can be obtained as induced subgraphs by employing a technique parallel to the dismantling of certain association schemes. Specific examples thus arise from (partial) spreads in some small generalized quadrangles.
Finally (if time permits), we offer a full classification of two-eigenvalue gain graphs with degree at most 4, or with a multiplicity at most 3.
Science Park 904, room D1.116 and online.