Arc-transitive digraphs of given out-valency and with blocks of given size

TL;DR

This paper constructs specific arc-transitive graphs and digraphs with prescribed valency and block size, advancing understanding of symmetry properties in graph theory.

Contribution

It introduces constructions of arc-transitive graphs and digraphs with specified out-valency and block size, expanding the class of known symmetric structures.

Findings

Constructed G-arc-transitive graphs with valency k and blocks of size m.

Constructed L-arc-transitive digraphs with out-valency k and blocks of size m.

Demonstrated the existence of such structures for given parameters.

Abstract

Given integers $k$ and $m$, we construct a $G$-arc-transitive graph of valency $k$ and an $L$-arc-transitive oriented digraph of out-valency $k$ such that $G$ and $L$ both admit blocks of imprimitivity of size $m$.