On the clique number of integral circulant graphs

TL;DR

This paper investigates the clique number of integral circulant graphs, confirming a conjecture for cases with one or two divisors and providing counterexamples for cases with three or more divisors.

Contribution

It fully solves the clique number problem for graphs with up to two divisors and disproves the conjecture for graphs with three or more divisors.

Findings

Clique number divides the number of vertices for graphs with one or two divisors.

Counterexamples show the conjecture does not hold for graphs with three or more divisors.

Complete characterization of clique numbers for certain classes of integral circulant graphs.

Abstract

The concept of gcd-graphs is introduced by Klotz and Sander, which arises as a generalization of unitary Cayley graphs. The gcd-graph $X_n (d_1,...,d_k)$ has vertices $0,1,...,n-1$, and two vertices $x$ and $y$ are adjacent iff $\gcd(x-y,n)\in D = \{d_1,d_2,...,d_k\}$. These graphs are exactly the same as circulant graphs with integral eigenvalues characterized by So. In this paper we deal with the clique number of integral circulant graphs and investigate the conjecture proposed in \cite{klotz07} that clique number divides the number of vertices in the graph $X_n (D)$. We completely solve the problem of finding clique number for integral circulant graphs with exactly one and two divisors. For $k \geqslant 3$, we construct a family of counterexamples and disprove the conjecture in this case.