Symmetrizable integer matrices having all their eigenvalues in the interval [-2,2]
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 [-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 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).

Received: 2019-10-22
Accepted: 2020-02-11
Revised after acceptance: 2020-03-18
Published online: 2020-06-02
DOI: https://doi.org/10.5802/alco.113
Classification: 15A18,  15B36,  11C20
Keywords: Symmetrizable matrices, spectral radius, Dynkin diagrams
@article{ALCO_2020__3_3_775_0,
     author = {McKee, James and Smyth, Chris},
     title = {Symmetrizable integer matrices having all their eigenvalues in the interval $[-2,2]$},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {3},
     number = {3},
     year = {2020},
     pages = {775-789},
     doi = {10.5802/alco.113},
     language = {en},
     url = {alco.centre-mersenne.org/item/ALCO_2020__3_3_775_0/}
}
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/item/ALCO_2020__3_3_775_0/

[1] Arnold, Vladimir Problems in present day mathematics, Mathematical developments arising from Hilbert problems. Proceedings of the symposium in pure mathematics of the American Mathematical Society, held at Northern Illinois University, DeKalb, Illinois, May 1974. (Proc. Symp. Pure Math.) Volume 28 (1976) (p. 46, in Felix E. Browder: “Problems in present day mathematics”, Problem VIII. The A-D-E classifications (V. Arnold)) | Zbl 0326.00002

[2] Carter, Roger W. Lie algebras of finite and affine type, Cambridge Studies in Advanced Mathematics, Volume 96, Cambridge University Press, Cambridge, 2005, xviii+632 pages | Article | MR 2188930 | Zbl 1110.17001

[3] Fisk, Steve A very short proof of Cauchy’s interlace theorem for eigenvalues of hermitian matrices, Amer. Math. Monthly, Volume 112 (2005) no. 2 (p. 118, see also https://arxiv.org/abs/math/0502408v1)

[4] Greaves, Gary Cyclotomic matrices over real quadratic integer rings, Linear Algebra Appl., Volume 437 (2012) no. 9, pp. 2252-2261 | Article | MR 2954487 | Zbl 1250.15031

[5] Hazewinkel, Michel; Hesselink, Wim; Siersma, Dirk; Veldkamp, Ferdinand D. The ubiquity of Coxeter–Dynkin diagrams (an introduction to the A-D-E problem), Nieuw Arch. Wisk. (3), Volume 25 (1977) no. 3, pp. 257-307 | MR 491673 | Zbl 0377.20037

[6] Kouachi, Said The Cauchy interlace theorem for symmetrizable matrices (2016) (https://arxiv.org/abs/1603.04151)

[7] McKee, James; Smyth, Chris Integer symmetric matrices having all their eigenvalues in the interval [-2,2], J. Algebra, Volume 317 (2007) no. 1, pp. 260-290 | Article | MR 2360149 | Zbl 1140.15007

[8] McKee, James; Smyth, Chris Symmetrizable matrices, quotients, and the trace problem (2020) (in preparation)

[9] Smith, John H. Some properties of the spectrum of a graph, Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969) (1970), pp. 403-406 | MR 0266799 | Zbl 0249.05136

[10] Sternberg, Shlomo Lie Algebras, 2004 (Harvard University, http://people.math.harvard.edu/~shlomo/docs/lie_algebras.pdf)