## Cyclotomic matrices and graphs

##### 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. Six rings OQ(
p
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.