# New results on the degree/diameter problem of mixed Abelian Cayley graphs

## TL;DR

This paper explores the degree/diameter problem for mixed Abelian Cayley graphs, presenting new families of graphs with large vertex counts for fixed degrees and diameters, some of which are proven optimal.

## Contribution

It introduces a generalized approach to construct mixed Abelian Cayley graphs with large order, extending the understanding of their degree/diameter properties.

## Key findings

- Constructed families of mixed Abelian Cayley graphs with asymptotically large size
- Some of the graphs' parameters are proven to be optimal
- Extended the algebraic framework for analyzing these graphs

## Abstract

Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case in which such graphs are Cayley graphs of Abelian groups. These groups can be constructed by using a generalization to $\mathbb{Z}^n$ of the concept of congruence in $\mathbb{Z}$. Here we use this approach to present some families of mixed graphs, which, for every fixed value of the degree, have an asymptotically large number of vertices as the diameter increases. In some cases, the results obtained are shown to be optimal.