On the non-existence of Abelian Moore Cayley graphs with excess one
Wei He · Discrete Mathematics Letters · 2021
The order of an Abelian Cayley graph of degree 2n and diameter 2 cannot exceed 2n 2 + 2n + 1, which is the famous Abelian Cayley-Moore bound.Leung and Zhou [J.Combin.Theory Ser.A 171 (2020) Art# 105157] recently shown that such a graph attaining the aforementioned bound exists if and only if n = 1, 2. This note is concerned with the Abelian Cayley graphs of degree 2n and diameter 3, whose order is 2n 2 + 2n + 2, one larger than the Abelian Cayley-Moore bound of degree 2n and diameter 2. Their generating set denoted by S satisfies | S2 | = 2n 2 + 2n + 1 where S = S ∪ {e}, e being the identity element of the underlying group.For n = 1, 2, it is easy to find examples.For n > 2, several non-existence results for infinitely many values of n are provided by using two methods, which are related to symmetric polynomials theories and algebraic number theory.