Maximal diameter of integral circulant graphs

Abstract

Integral circulant graphs are proposed as models for quantum spin networks enabling perfect state transfer. Understanding the potential information transfer between nodes in such networks involves calculating the maximal graph diameter. The integral circulant graph ${\mathrm{ICG}}_n\left(D\right)$ has vertex set $Z_n=\{0,1,2,\dots ,n-1\}$, with vertices a and b adjacent if $\gcd (a-b,n)\in D$, where $D\subseteq \{d:d|n,1\le d<n\}$. Building on the upper bound $2\left|D\right|+1$ for the diameter provided by Saxena, Severini, and Shparlinski, we prove that the maximal diameter of ${\mathrm{ICG}}_n\left(D\right)$ for a given order n with prime factorization $p_1^{{\alpha }_1}\cdots p_k^{{\alpha }_k}$ is $r\left(n\right)$ or $r\left(n\right)+1$, where $r\left(n\right)=k+\left|\right\{i|{\alpha }_i>1,1\le i\le k\}|$. We show that a divisor set D with $\left|D\right|\le k$ achieves this bound. We calculate the maximal diameter for graphs of order n and divisor set cardinality $t\le k$, identifying all extremal graphs and improving the previous upper bound.