Since the introduction of the Hermitian adjacency matrix for digraphs, interest in so-called complex unit gain graphs has surged. In this talk, we consider gain graphs with two distinct eigenvalues. Analogously to graphs with 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 related to lines in complex space with few angles, including 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 association schemes. Specific examples thus arise from (partial) spreads in some small generalized quadrangles. Finally, we offer a full classification of two-eigenvalue gain graphs with degree at most 4, or with multiplicity at most 3.

(Joint work with Pepijn Wissing)