Integer Symmetric Matrices of Small Spectral Radius and Small Mahler Measure

In a previous paper, we completely described cyclotomic matrices—integer symmetric matrices of spectral radius at most 2. In this paper, we find all minimal noncyclotomic matrices. As a consequence, we are able to determine all integer symmetric matrices of spectral radius at most 2.019, and to determine all integer symmetric matrices whose Mahler measure is at most 1.3. In particular, we solve the strong version of Lehmer’s problem for integer symmetric matrices: all noncyclotomic matrices have Mahler measure at least ‘Lehmer’s number’ λ0=1.17628….