Vertex-transitive graphs and their arc-types

TL;DR

This paper investigates the arc-types of finite vertex-transitive graphs, characterizes their structure, and demonstrates that most partitions of valency are realizable as arc-types, including their behavior under Cartesian products.

Contribution

It introduces a detailed classification of arc-types for vertex-transitive graphs and proves that all but two specific partitions are realizable, expanding understanding of graph symmetries.

Findings

Determined arc-types for several graph families.

Proved arc-type of Cartesian product is the sum of component arc-types.

Most partitions of valency are realizable as arc-types, except for two cases.

Abstract

Let $X$ be a finite vertex-transitive graph of valency $d$, and let $A$ be the full automorphism group of $X$. Then the arc-type of $X$ is defined in terms of the sizes of the orbits of the action of the stabiliser $A_v$ of a given vertex $v$ on the set of arcs incident with $v$. Specifically, the arc-type is the partition of $d$ as the sum $$n_1 + n_2 + \dots + n_t + (m_1 + m_1) + (m_2 + m_2) + \dots + (m_s + m_s),$$ where $n_1, n_2, \dots, n_t$ are the sizes of the self-paired orbits, and $m_1,m_1, m_2,m_2, \dots, m_s,m_s$ are the sizes of the non-self-paired orbits, in descending order. In this paper, we find the arc-types of several families of graphs. Also we show that the arc-type of a Cartesian product of two `relatively prime' graphs is the natural sum of their arc-types. Then using these observations, we show that with the exception of $1+1$ and $(1+1)$, every partition as defined above is realisable, in the sense that there exists at least one graph with the given partition as its arc-type.