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Cyclotomic matrices and graphs over the ring of integers of some imaginary quadratic fields. (English) Zbl 1238.05166

Let \(A\) be an \(n \times n\) hermitian matrix over the ring of integers of an imaginary quadratic number field and let \(\chi_A(x) \in \mathbb{Z}[x]\) be its characteristic polynomial. Then \(A\) is called cyclotomic if \(x^n \chi_A(x+1/x)\) is a cyclotomic polynomial (i.e., all roots are roots of unity), or equivalently, all eigenvalues are in the interval \([-2,2]\). The author points out that for square-free \(d \notin \{-1,-2,-3,-7,-11,-15\}\), all hermitian cyclotomic matrices over the ring of integers of \(\mathbb{Q}(\sqrt{d})\) have integer entries, hence are symmetric integer cyclotomic matrices, whose classification was achieved by J. McKee and C. Smyth [“Integer symmetric matrices having all their eigenvalues in the interval \([-2,2]\),” J. Algebra 317, No. 1, 260–290 (2007; Zbl 1140.15007)]. The author gives a classification of all hermitian cyclotomic matrices over the ring of integers of \(\mathbb{Q}(\sqrt{d})\) for \(d \in \{-2,-7,-11,-15\}\). Thus the only remaining cases are \(d \in \{ -1, -3 \}\).
Extending the work of J. H. Smith [“Some properties of the spectrum of a graph,” Combinat. Struct. Appl., Proc. Calgary Int. Conf. Comb. Struct. Appl., Calgary 1969, 403–406 (1970; Zbl 0249.05136)] (developed by McKee and Smyth [loc. cit.]), the author associates to each indecomposable hermitian cyclotomic matrix a cyclotomic \(\mathcal{L}\)-signed charged graph (or \(\mathcal{L}\)-graph), where \(\mathcal{L}\) is the set of integers of \(\mathbb{Q}(\sqrt{d})\) with norm in the interval \([0,4]\). The matrix is viewed as a generalized adjacency matrix of the graph. Maximal such cyclotomic \(\mathcal{L}\)-graphs are then classified and it is shown that any such graph embeds into a maximal one. For \(d \in \{ -11, -15\}\), it is proved that any maximal cyclotomic \(\mathcal{L}\)-graph, not already appearing in the above classifications (i.e., arising from symmetric integer matrices), is equivalent to one of a finite number of sporadic graphs.
The main step in the classification for \(d \in \{-2, -7\}\) is that a cyclotomic \(\mathcal{L}\)-graph is maximal if and only if it is 4-cyclotomic, i.e., each vertex has weight 4. In the latter cases, aside from a finite number of sporadic examples, there appear three new infinite families of maximal cyclotomic \(\mathcal{L}\)-graphs in the classification. The author’s introduction includes a clear exposition of the relationship between such matrix classification results and Lehmer’s conjecture.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
11R11 Quadratic extensions
15B57 Hermitian, skew-Hermitian, and related matrices
15B36 Matrices of integers
15B33 Matrices over special rings (quaternions, finite fields, etc.)
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
05C22 Signed and weighted graphs

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References:

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