An improved Moore bound and some new optimal families of mixed Abelian Cayley graphs

TL;DR

This paper improves Moore bounds for mixed Abelian Cayley graphs by leveraging element order details, leading to new large families of graphs with optimal cases identified.

Contribution

It introduces improved bounds for mixed Abelian Cayley graphs and constructs new families with asymptotically large size, some proven to be optimal.

Findings

Improved Moore bounds for mixed Abelian Cayley graphs.

Construction of new large families of such graphs.

Identification of cases where bounds are optimal.

Abstract

We consider the case in which mixed graphs (with both directed and undirected edges) are Cayley graphs of Abelian groups. In this case, some Moore bounds were derived for the maximum number of vertices that such graphs can attain. We first show these bounds can be improved if we know more details about the order of some elements of the generating set. Based on these improvements, we present some new families of mixed graphs. For every fixed value of the degree, these families have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are shown to be optimal.