On $σ$-arithmetic graphs of finite groups

Let $G$ be a finite group and $σ$ a partition of the set of all? primes $\Bbb{P}$, that is, $σ=\{σ_i \mid i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} σ_i$ and $σ_i\cap σ_j= \emptyset$ for all $i\ne j$. If $n$ is an integer, we write $σ(n)=\{σ_i \mid σ_{i}\cap π(n)\ne \emptyset \}$ and $σ(G)=σ(|G|)$. We call a graph $Γ$ with the set of all vertices $V(Γ)=σ(G)$ ($G\ne 1$) a $σ$-arithmetic graph of $G$, and we associate with $G\ne 1$ the following three directed $σ$-arithmetic graphs: (1) the $σ$-Hawkes graph $Γ_{Hσ}(G)$ of $G$ is a $σ$-arithmetic graph of $G$ in which $(σ_i, σ_j)\in E(Γ_{Hσ}(G))$ if $σ_j\in σ(G/F_{\{σ_i\}}(G))$; (2) the $σ$-Hall graph $Γ_{σHal}(G)$ of $G$ in which $(σ_i, σ_j)\in E(Γ_{σHal}(G))$ if for some Hall $σ_i$-subgroup $H$ of $G$ we have $σ_j\in σ(N_{G}(H)/HC_{G}(H))$; (3) the $σ$-Vasil'ev-Murashko graph $Γ_{{\mathfrak{N}_σ}}(G)$ of $G$ in which $(σ_i, σ_j)\in E(Γ_{{\mathfrak{N}_σ}}(G))$ if for some ${\mathfrak{N}_{σ}}$-critical subgroup $H$ of $G$ we have $σ_i \in σ(H)$ and $σ_j\in σ(H/F_{\{σ_i\}}(H))$. In this paper, we study the structure of $G$ depending on the properties of these three graphs of $G$.

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