Maximal diameter of integral circulant graphs

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 ICG(n)(D) has vertex set Z(n) = {0, 1, 2,..., n - 1}, with vertices a and b adjacent if gcd(a - b, n) is an element of D, where D subset of {d : d vertical bar n, 1 <= d < n}. Building on the upper bound 2 vertical bar D vertical bar + 1 for the diameter provided by Saxena, Severini, and Shparlinski, we prove that the maximal diameter of ICG(n)(D) for a given order nwith prime factorization p(1)(alpha 1) center dot center dot center dot p(alpha)(alpha k) is r(n) or r(n) + 1, where r(n) = k +vertical bar{i vertical bar alpha(i) > 1, 1 <= i <= k}vertical bar. We show that a divisor set Dwith vertical bar D vertical bar <= k achieves this bound. We calculate the maximal diameter for graphs of order nand divisor set cardinality t = k, identifying all extremal graphs and improving the previous upper bound. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.