Two-arc-transitive bicirculants

TL;DR

This paper classifies all finite 2-arc-transitive bicirculant graphs, providing a comprehensive list of such graphs including well-known structures and new families, advancing understanding in algebraic graph theory.

Contribution

The paper offers a complete classification of finite 2-arc-transitive bicirculants, identifying all such graphs and their structural properties, which was previously unknown.

Findings

Identified all connected 2-arc-transitive bicirculant graphs.

Included classical graphs like Petersen, Desargues, and dodecahedron.

Described new families and characterized their parameters.

Abstract

In this paper, we determine the class of finite 2-arc-transitive bicirculants. We show that a connected $2$-arc-transitive bicirculant is one of the following graphs: $C_{2n}$ where $n\geqslant 2$, $\K_{2n}$ where $n\geqslant 2$, $\K_{n,n}$ where $n\geqslant 3$, $ \K_{n,n}-n\K_2$ where $n\geqslant 4$, $B(\PG(d-1,q))$ and $B'(\PG(d-1,q))$ where $d\geq 3$ and $q$ is a prime power, $ X_1(4,q)$ where $q\equiv 3\pmod{4}$ is a prime power, $\K_{q+1}^{2d}$ where $q$ is an odd prime power and $d\geq 2$ dividing $q-1$, $ AT_Q(1+q,2d)$ where $d\mid q-1$ and $d\nmid \frac{1}{2}(q-1)$, $ AT_D(1+q,2d)$ where $d\mid \frac{1}{2}(q-1)$ and $d\geq 2$, $\Gamma(d, q, r)$, where $d\geq 2$, $q$ is a prime power and $r|q-1$, Petersen graph, Desargues graph, dodecahedron graph, folded $5$-cube, $X(3,2)$, $ X_2(3)$, $ AT_Q(4,12)$, $GP(12,5)$, $GP(24,5)$, $B(H(11))$, $B'(H(11))$, $ AT_D(4,6)$ and $ AT_D(5,6)$.