Maximal diameter on a class of circulant graphs

Abstract

Integral circulant graphs are proposed as models for quantum spin networks. Specifically, it is important to know how far information can potentially be transferred between nodes of the quantum networks modeled by integral circulant graphs and this task is related to calculating the maximal diameter of a graph. The integral circulant graph (Formula Presented) has the vertex set (Formula Presented) and vertices a and b are adjacent if (Formula Presented), where (Formula Presented). Motivated by the result on the upper bound of the diameter of (Formula Presented) given in [N. Saxena, S. Severini, I. Shparlinski, Parameters of integral circulant graphs and periodic quantum dynamics, International Journal of Quantum Information 5 (2007), 417–430], which is equal to (Formula Presented), in this paper we prove that the maximal value of the diameter of the integral circulant graph (Formula Presented) of a given order n with its prime factorization (Formula Presented) and (Formula Presented), is equal to (Formula Presented). This way we further improve the upper bound of Saxena, Severini and Shparlinski. Moreover, we characterize all such extremal graphs. We also show that the upper bound is attainable for integral circulant graphs (Formula Presented) such that (Formula Presented).