**Symmetrizable integer matrices having all their eigenvalues in the interval [-2,2].** / Smyth, Chris; McKee, James.

Research output: Contribution to journal › Article

Forthcoming

**Symmetrizable integer matrices having all their eigenvalues in the interval [-2,2].** / Smyth, Chris; McKee, James.

Research output: Contribution to journal › Article

Smyth, C & McKee, J 2020, 'Symmetrizable integer matrices having all their eigenvalues in the interval [-2,2]', *Algebraic Combinatorics*.

Smyth, C., & McKee, J. (Accepted/In press). Symmetrizable integer matrices having all their eigenvalues in the interval [-2,2]. *Algebraic Combinatorics*.

Smyth C, McKee J. Symmetrizable integer matrices having all their eigenvalues in the interval [-2,2]. Algebraic Combinatorics. 2020 Feb 11.

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title = "Symmetrizable integer matrices having all their eigenvalues in the interval [-2,2]",

abstract = "The adjacency matrices of graphs form a special subset of the set of all integer symmetric matrices. The description of which graphs have all their eigenvalues in the interval [-2,2] (i.e., those having spectral radius at most 2) has been known for several decades. In 2007 we extended this classification to arbitrary integer symmetric matrices.In this paper we turn our attention to symmetrizable matrices. We classify the connected nonsymmetric but symmetrizable matrices which have entries in Z that are maximal with respect to having all their eigenvalues in [-2,2]. This includes a spectral characterisation of the affine and finite Dynkin diagrams that are not simply laced (much as the graph result gives a spectral characterisation of the simply laced ones). ",

keywords = "symmetrizable matrices, spectral radius, Dynkin diagrams",

author = "Chris Smyth and James McKee",

year = "2020",

month = feb

day = "11",

language = "English",

journal = "Algebraic Combinatorics",

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AU - McKee, James

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N2 - The adjacency matrices of graphs form a special subset of the set of all integer symmetric matrices. The description of which graphs have all their eigenvalues in the interval [-2,2] (i.e., those having spectral radius at most 2) has been known for several decades. In 2007 we extended this classification to arbitrary integer symmetric matrices.In this paper we turn our attention to symmetrizable matrices. We classify the connected nonsymmetric but symmetrizable matrices which have entries in Z that are maximal with respect to having all their eigenvalues in [-2,2]. This includes a spectral characterisation of the affine and finite Dynkin diagrams that are not simply laced (much as the graph result gives a spectral characterisation of the simply laced ones).

AB - The adjacency matrices of graphs form a special subset of the set of all integer symmetric matrices. The description of which graphs have all their eigenvalues in the interval [-2,2] (i.e., those having spectral radius at most 2) has been known for several decades. In 2007 we extended this classification to arbitrary integer symmetric matrices.In this paper we turn our attention to symmetrizable matrices. We classify the connected nonsymmetric but symmetrizable matrices which have entries in Z that are maximal with respect to having all their eigenvalues in [-2,2]. This includes a spectral characterisation of the affine and finite Dynkin diagrams that are not simply laced (much as the graph result gives a spectral characterisation of the simply laced ones).

KW - symmetrizable matrices, spectral radius, Dynkin diagrams

M3 - Article

JO - Algebraic Combinatorics

JF - Algebraic Combinatorics

SN - 2589-5486

ER -