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 $ICG_n (D)$ has the vertex set $Z_n = \{0, 1, 2, \ldots, n - 1\}$ and vertices $a$ and $b$ are adjacent if $\gcd(a-b,n)\in D$, where $D \subseteq \{d : d \mid n,\ 1\leq d<n\}$. Motivated by the result on the upper bound of the diameter of $ICG_n(D)$ 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 $ICG_n(D)$ of a given order $n$ with its prime factorization $p_1^{\alpha_1}\cdots p_k^{\alpha_k}$, is equal to $r(n)$ or $r(n)+1$, where $r(n)=k + |\{ i \ | \alpha_i> 1,\ 1\leq i\leq k \}|$, depending on whether $n\not\in 4N+2$ or not, respectively. Furthermore, we show that, for a given order $n$, a divisor set $D$ with $|D|\leq k$ 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 $t\leq k$ 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.

翻译：整循环图被提出作为量子自旋网络的模型，该网络允许一种称为完美态转移的量子现象。具体而言，了解由整循环图建模的量子网络中节点之间信息可能传输的最大距离至关重要，而这与计算图的最大直径有关。整循环图 $ICG_n (D)$ 的顶点集为 $Z_n = \{0, 1, 2, \ldots, n - 1\}$，顶点 $a$ 和 $b$ 相邻当且仅当 $\gcd(a-b,n)\in D$，其中 $D \subseteq \{d : d \mid n,\ 1\leq d<n\}$。受关于 $ICG_n(D)$ 直径上界结果（见 [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]）的启发，该上界为 $2|D|+1$，在本文中我们证明，对于给定阶数 $n$（其素因子分解为 $p_1^{\alpha_1}\cdots p_k^{\alpha_k}$），整循环图 $ICG_n(D)$ 直径的最大值等于 $r(n)$ 或 $r(n)+1$，其中 $r(n)=k + |\{ i \ |\alpha_i> 1,\ 1\leq i\leq k \}|$，具体取决于 $n$ 是否属于 $4N+2$。此外，我们证明对于给定的阶数 $n$，总可以找到一个满足 $|D|\leq k$ 的除数集 $D$，使得该上界可达。最后，我们计算了给定阶数 $n$ 以及除数集基数 $t\leq k$ 的整循环图类中的最大直径，并刻画了所有极值图。实际上，我们证明了最大直径可能取值为 $2t$、$2t+1$、$r(n)$ 和 $r(n)+1$，具体取决于 $t$ 和 $n$ 的值。由此，我们进一步改进了 Saxena、Severini 和 Shparlinski 的上界，并刻画了所有直径等于 $2|D|+1$ 的图，从而推广了该论文中的一个结果。