Integer symmetric matrices of small spectral radius and small Mahler measure (0907.0371v1)

Abstract: 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" 1.17628... .