Cayley graphs of given degree and diameter for cyclic, Abelian, and metacyclic groups

Macbeth, Heather; Šiagiová, Jana and Širáň, Jozef (2012). Cayley graphs of given degree and diameter for cyclic, Abelian, and metacyclic groups. Discrete Mathematics, 312(1) pp. 94–99.

DOI: https://doi.org/10.1016/j.disc.2011.03.038

Abstract

Let CC(d,2) and AC(d,2) be the largest order of a Cayley graph of a cyclic and an Abelian group, respectively, of diameter 2 and a given degree d. There is an obvious upper bound of the form CC(d,2)≤AC(d,2)≤d²/2+d+1. We prove a number of lower bounds on both quantities for certain infinite sequences of degrees d related to primes and prime powers, the best being CC(d,2)≥(9/25)(d+3)(d−2) and AC(d,2)≥(3/8)(d²−4). We also offer a result for Cayley graphs of metacyclic groups for general degree and diameter.

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