The Geometric Kernel of Integral Circulant Graphs

Abstract

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 \emph{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 $\mathrm{ICG}(n,\mathcal{D})$ 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 $\mathrm{ICG}(n,\mathcal{D})$.

Graphs Defined on Rings: A Review

“…Appropriate representation of the circulant graphs on a Euclidean plane unveils the rotational symmetry of the graph. As previously known, unitary Cayley graphs are integral circulant graphs and, therefore, such a suitable representation or drawing, called the unit circle drawing of a unitary Cayley graph, was examined in [84]. The unit circle drawing of the graph X n is simply a drawing of the graph X n such that the vertices are placed equidistantly on a unit circle on the complex plane C and the edges are drawn as line segments.…”

“…It was proven in [85] that the central hole in the unit circle drawing of any circulant graph on n > 3 vertices is a regular n-gon. Therefore, only the size of the geometric kernel for X n , which is already known to be an n-gon, had to be determined, in [84], by computing the kernel radius, given by the formula max{k : 1 ≤ k < n 2 , gcd(k, n) = 1}. Only integers less than n 2 were considered because there is no central hole when the edge (k, k 2 ) exists in the unit circle drawing of a graph.…”

Quantum state transfer on integral oriented circulant graphs

“…M. Bašić and A. Ilić determined the clique number [4], the automorphism group [5], and the chromatic number [24] of integral circulant graphs. Sander [32][33][34] studied some structural properties of integral circulant graphs.…”