# ALGEBRAIC COMBINATORICS

Symmetrizable integer matrices having all their eigenvalues in the interval $\left[-2,2\right]$
Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 775-789.

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 $\left[-2,2\right]$ (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 $\left[-2,2\right]$. 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).

Accepted:
Revised after acceptance:
Published online:
DOI: 10.5802/alco.113
Classification: 15A18,  15B36,  11C20
Keywords: Symmetrizable matrices, spectral radius, Dynkin diagrams
McKee, James 1; Smyth, Chris 2

1 Department of Mathematics Royal Holloway University of London Egham Hill, Egham, Surrey, TW20 0EX, U.K.
2 School of Mathematics University of Edinburgh Edinburgh EH9 3FD Scotland, U.K.
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McKee, James; Smyth, Chris. Symmetrizable integer matrices having all their eigenvalues in the interval $[-2,2]$. Algebraic Combinatorics, Volume 3 (2020) no. 3, pp. 775-789. doi : 10.5802/alco.113. https://alco.centre-mersenne.org/articles/10.5802/alco.113/

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