Unit gain graphs with two distinct eigenvalue and systems of lines in complex space

TL;DR

This paper classifies complex unit gain graphs with exactly two eigenvalues, revealing their structural symmetry and connection to systems of lines in complex space, with applications to quantum information and computational methods.

Contribution

It provides a complete classification of two-eigenvalue gain graphs with degree at most 4 or multiplicity at most 3, linking spectral graph theory to complex line systems and quantum geometries.

Findings

Classified all two-eigenvalue gain graphs with degree ≤ 4 or multiplicity ≤ 3.

Established a correspondence between such graphs and systems of lines like SIC-POVMs and MUBs.

Used simulated annealing to computationally find examples of these graphs.

Abstract

Since the introduction of the Hermitian adjacency matrix for digraphs, interest in so-called complex unit gain graphs has surged. In this work, we consider gain graphs whose spectra contain the minimum number of two distinct eigenvalues. Analogously to graphs with few distinct eigenvalues, a great deal of structural symmetry is required for a gain graph to attain this minimum. This allows us to draw a surprising parallel to well-studied systems of lines in complex space, through a natural correspondence to unit-norm tight frames. We offer a full classification of two-eigenvalue gain graphs with degree at most $4$, or with multiplicity at most $3$. Intermediate results include an extensive review of various relevant concepts related to lines in complex space, including SIC-POVMs, MUBs and geometries such as the Coxeter-Todd lattice, and many examples obtained as induced subgraphs by employing a technique parallel to the dismantling of association schemes. Finally, we touch on an innovative application of simulated annealing to find examples by computer.