Integral circulant graphs are proposed as models for quantum spin networks
that permit a quantum phenomenon called perfect state transfer. Specifically,
it is important to know how far information can potentially be transferred
between nodes of the quantum networks modelled by integral circulant graphs and
this task is related to calculating the maximal diameter of a graph. The
integral circulant graph ICGn(D) has the vertex set Zn={0,1,2,…,n−1} and vertices a and b are adjacent if gcd(a−b,n)∈D,
where D⊆{d:d∣n,1≤d<n}. Motivated by the result on
the upper bound of the diameter of ICGn(D) given in [N. Saxena, S. Severini,
I. Shparlinski, \textit{Parameters of integral circulant graphs and periodic
quantum dynamics}, International Journal of Quantum Information 5 (2007),
417--430], according to which 2∣D∣+1 represents one such bound, in this paper
we prove that the maximal value of the diameter of the integral circulant graph
ICGn(D) of a given order n with its prime factorization
p1α1⋯pkαk, is equal to r(n) or r(n)+1, where
r(n)=k+∣{i∣αi>1,1≤i≤k}∣, depending on whether
n∈4N+2 or not, respectively. Furthermore, we show that, for a given
order n, a divisor set D with ∣D∣≤k can always be found such that
this bound is attained. Finally, we calculate the maximal diameter in the class
of integral circulant graphs of a given order n and cardinality of the
divisor set t≤k and characterize all extremal graphs. We actually show
that the maximal diameter can have the values 2t, 2t+1, r(n) and r(n)+1
depending on the values of t and n. This way we further improve the upper
bound of Saxena, Severini and Shparlinski and we also characterize all graphs
whose diameters are equal to 2∣D∣+1, thus generalizing a result in that
paper.Comment: 29 pages, 1 figur