## 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

### Citations:

Zbl 1140.15007; Zbl 0249.05136

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### References:

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