The geometric kernel of integral circulant graphs (Q2048568)

Summary: By a suitable representation in the Euclidean plane, each circulant graph \(G\), i.e. a graph with a circulant adjacency matrix \({\mathcal A}(G)\), reveals its rotational symmetry and, as the drawing's most notable feature, a central hole, the so-called geometric kernel of \(G\). Every integral circulant graph \(G\) on \(n\) vertices, i.e. satisfying the additional property that all of the eigenvalues of \({\mathcal A}(G)\) are integral, is isomorphic to some graph \(\text{ICG}(n,\mathcal{D})\) having vertex set \(\mathbb{Z}/n\mathbb{Z}\) and edge set \(\{\{a,b\}:\, a,b\in\mathbb{Z}/n\mathbb{Z} ,\, \gcd(a-b,n)\in \mathcal{D}\}\) for a uniquely determined set \(\mathcal{D}\) of positive divisors of \(n\). A lot of recent research has revolved around the interrelation between graph-theoretical, algebraic and arithmetic properties of such graphs. In this article we examine arithmetic implications imposed on \(n\) by a geometric feature, namely the size of the geometric kernel of \(\text{ICG}(n,\mathcal{D})\).