$2$-arc-transitive graphs of order $kp^n$

Gabriel Verret
The University of Western Australia



Content: Over the past few years, there has been an explosion of results classifying arc-transitive (or $2$-arc-transitive, arc-regular, etc...) graphs of valency $d$ and order $kp^n$, where $d,k,n$ are fixed integers and $p$ is a prime that is allowed to vary. These results are often proved by standard methods and thus there has been some recent efforts to unify the results by proving more general ones. For example, a recent theorem of Conder, Li and Poto\v{c}nik shows that, for \emph{every} fixed $k$ and $d\geq 3$, there exists only finitely many $2$-arc-transitive graphs of valency $d$ and order $kp$, for $p$ a prime. They also prove something similar for graphs of order $kp^2$. Recently, we have proved an analog of their result for $n\geq 3$. More precisely, we have shown that there exists a function $g$ such that there are only finitely many $d$-valent 2-arc-transitive graphs of order $kp^n$ with $d>g(n)$ and $p$ a prime. This is joint work with Luke Morgan and Eric Swartz.

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