Jordan form of coxeter transformations and applications to representations of finite graphs

Subbotin, V. F.; Stekolshchik, Rafael

doi:10.1007/bf01077574

Cited by 12 publications

(6 citation statements)

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“…We also prove the following theorem concerning joins of trees. [31] if the common root z of the polynomials cox T 1 (t), . .…”

Section: Resultsmentioning

confidence: 99%

Coxeter polynomials of Salem trees

Evripidou

2015

Colloq. Math.

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Abstract. We compute the Coxeter polynomial of a family of Salem trees and also the limit of the spectral radius of their Coxeter transformations as the number of their vertices tends to infinity. We also prove that if z is a root of multiplicities m 1 , . . . , m k for the Coxeter polynomials of the trees T 1 , . . . , T k , then z is a root for the Coxeter polynomial of their join, of multiplicity at least min{m − m 1 , . . . , m − m k } where m = m 1 + . . . + m k .

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“…We also prove the following theorem concerning joins of trees. [31] if the common root z of the polynomials cox T 1 (t), . .…”

Section: Resultsmentioning

confidence: 99%

Coxeter polynomials of Salem trees

Evripidou

2015

Colloq. Math.

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“…This is, of course, also trivially true when C defines a Dynkin graph (for, c is of finite order). In the case that C defines a tree, sharper results on the eigenvalues of the corresponding Coxeter transformations were obtained by N. A' Campo [1] and V. F. Subbotin and R. B. Stekolscik [7]; however, they do not generalize to the general situation of nonsymmetrizable matrices.…”

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confidence: 95%

Eigenvalues of Coxeter transformations and the Gel\cprime fand-Kirillov dimension of the preprojective algebras

Dlab¹,

Ringel²

1981

Proc. Amer. Math. Soc.

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“…The properties of CT's are well-studied [1][2][3][4] for all connected Coxeter graphs with B Γ (x) > 0 (Dynkin graphs) and with B Γ (x) ≥ 0 (affine (completed) Dynkin graphs) except for the affine cycle A n (simple cycle). In [1] (P. 146) it is proved that all CT's for graphs without cycles, independently of the choice of bijection (1), are conjugate in the group W .…”

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confidence: 99%

“…Therefore, their spectral properties coincide. For the graph A n , it is not the case, which gives no possibility to transfer completely the results from [1][2][3][4].…”

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confidence: 99%