Abstract
Given a circulant graph Cay(Zn,S), we precisely determine its splitting field and algebraic degree, i.e. the least algebraic extension K|Q of the rationals, which contains all eigenvalues of Cay(Zn,S), and its respective degree [K:Q]. This generalizes a result of So who classified all integral circulant graphs. We prove that there is a deep connection between Schur rings and the splitting field of circulant graphs, which are subfields of cyclotomic fields, and solve the inverse Galois problem for circulant graphs showing that every finite abelian extension of the rationals is the splitting field of some circulant graph. Moreover, we deduce some new necessary criteria for isospectrality of circulant graphs and give a graph-theoretical interpretation of the algebraic degree of circulant graphs: we prove that the algebraic degree of a circulant graph is determined by its automorphism group.
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