Integer symmetric matrices of small spectral radius and small Mahler measure

TL;DR

This paper classifies integer symmetric matrices with small spectral radius and Mahler measure, solving a strong version of Lehmer's problem for such matrices by identifying minimal noncyclotomic cases.

Contribution

It identifies all minimal noncyclotomic matrices and determines bounds for spectral radius and Mahler measure for integer symmetric matrices.

Findings

All integer symmetric matrices with spectral radius ≤ 2.019 are classified.

Mahler measure for noncyclotomic matrices is at least Lehmer's number 1.17628.

The strong Lehmer's problem is solved for integer symmetric matrices.

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... .