A graph is called an integral graph if it has an integral spectrum i.e., all eigenvalues are integers. A graph is called circulant graph if it is Cayley graph on the circulant group, i.e., its adjacency matrix is circulant. The rank of a graph is defined to be the rank of its adjacency matrix. This importance of the rank, due to applications in physics, chemistry and combinatorics. In this paper, using Ramanujan sums, we study the rank of integral circulant graphs and gave some simple computational formulas for the rank and provide an example which shows the formula is sharp. 

A graph is called an integral graph if it has an integral spectrum i.e., all eigenvalues are integers. A graph is called circulant graph if it is Cayley graph on the circulant group, i.e., its adjacency matrix is circulant. The rank of a graph is defined to be the rank of its adjacency matrix. This importance of the rank, due to applications in physics, chemistry and combinatorics. In this paper, using Ramanujan sums, we study the rank of integral circulant graphs and gave some simple computational formulas for the rank and provide an example which shows the formula is sharp.