# Connectedness of two-sided group digraphs and graphs

## TL;DR

This paper characterizes the conditions for connectedness in two-sided group digraphs and graphs, addressing open problems and providing new insights into their structure using group actions.

## Contribution

It offers a comprehensive characterization of connectedness in two-sided group digraphs and graphs, solving four open problems and proposing five new ones.

## Key findings

- Characterized weak and strong connectedness conditions.
- Counted connected components explicitly.
- Addressed open problems on valency and connectedness.

## Abstract

Two-sided group digraphs and graphs, introduced by Iradmusa and Praeger, provide a generalization of Cayley digraphs and graphs in which arcs are determined by left and right multiplying by elements of two subsets of the group. We characterize when two-sided group digraphs and graphs are weakly and strongly connected and count connected components, using both an explicit elementary perspective and group actions. Our results and examples address four open problems posed by Iradmusa and Praeger that concern connectedness and valency. We pose five new open problems.