Today I successfully defended my PhD thesis, Cyclotomic Matrices and Graphs. There are of course numerous corrections to be made, but I hope to have those done within the next couple of weeks and to make the final version available online. Until then, here is the abstract:
We generalise the study of cyclotomic matrices – those with all eigenvalues in the interval [-2,2] – from symmetric rational integer matrices to Hermitian matrices with entries from rings of integers of imaginary quadratic fields. As in the rational integer case, a corresponding graph-like structure is defined.
We introduce the notion of `4-cyclotomic’ matrices and graphs, prove that they are necessarily maximal cyclotomic, and classify all such objects up to equivalence. The six rings of integers for Q(√(d)) for d=-1, -2, -3, -7, -11, -15 give rise to examples not found in the rational-integer case; in four (d=-1, -2, -3, -7) we recover infinite families as well as sporadic cases.
For d=-15, -11, -7, -2, we demonstrate that a maximal cyclotomic graph is necessarily 4-cyclotomic and thus the presented classification determines all cyclotomic matrices/graphs for those fields. For the same values of d we then identify the minimal noncyclotomic graphs and determine their Mahler measures; no such graph has Mahler measure less than 1.35 unless it admits a rational-integer representative.