Isomorphism testing for circulant graphs C_n(a,b)

In this paper we focus on connected directed/undirected circulant graphs $C_n(a,b)$. We investigate some topological characteristics, and define a simple combinatorial model, which is new for the topic. Building on such a model, we derive a necessary and sufficient condition to test whether two circulant graphs $C_n(a,b)$ and $C_n(a',b')$ are isomorphic or not. The method is entirely elementary and consists of comparing two suitably computed integers in $\{1, \dots, \frac{n}{\gcd(n,a)\gcd(n,b)}-1\}$, and of verifying if $\{\gcd(n,a),\gcd(n,b)\}=\{\gcd(n,a'),\gcd(n,b')\}$. It also allows for building the mapping function in linear time. In addition, properties of the classes of mutually isomorphic graphs are analyzed.