On some graphs associated with the finite alternating groups

Abstract

Let $P_0(A_n), \widetilde{P}_0(A_n), P_0(\mathcal{T}(A_n))$ and $\mathcal{O}_0(A_n)$ be respectively the proper power graph, the proper quotient power graph, the proper power type graph and the proper order graph of the alternating group $A_n$, for $n\geq 3.$ We determine the number of the components of those graphs. In particular, we prove that the power graph $P(A_n)$ is $2$-connected if and only if the power type graph $P(\mathcal{T}(A_n))$ is $2$-connected, if and only if either $n = 3$ or none of $n, n-1, n-2, \frac{n}{2}$ and ${\frac{n-1}{2}}$ is a prime. We also give some information on the properties of those components.