On $\sigma$-arithmetic graphs of finite groups

Let $G$ be a finite group and $\sigma$ a partition of the set of all? primes $\mathbb{P}$, that is, $\sigma =\{{\sigma }_i\mid i\in I\}$, where $\mathbb{P}={\bigcup }_{i\in I}{\sigma }_i$ and ${\sigma }_i\cap {\sigma }_j=\mathrm{\varnothing }$ for all $i\mathrm{\ne }j$. If $n$ is an integer, we write $\sigma(n)={\sigma_i \mid \sigma_{i}\cap \pi (n)\ne \emptyset }$and$\sigma (G)=\sigma (|G|)$. We call a graph $\mathrm{\Gamma }$ with the set of all vertices $V(\mathrm{\Gamma })=\sigma (G)$ ($G\mathrm{\ne }1$) a $\sigma$-arithmetic graph of $G$, and we associate with $G\mathrm{\ne }1$ the following three directed $\sigma$-arithmetic graphs: (1) the $\sigma$-Hawkes graph ${\mathrm{\Gamma }}_{H\sigma }(G)$ of $G$ is a $\sigma$-arithmetic graph of $G$ in which $({\sigma }_i,{\sigma }_j)\in E({\mathrm{\Gamma }}_{H\sigma }(G))$ if $\sigma_j\in \sigma (G/F_{{\sigma_i}}(G))$;(2)the$\sigma$-Hallgraph$\Gamma_{\sigma Hal}(G)$of$G$inwhich$(\sigma_i, \sigma_j)\in E(\Gamma_{\sigma Hal}(G))$ if for some Hall ${\sigma }_i$-subgroup $H$ of $G$ we have $\sigma_j\in \sigma (N_{G}(H)/HC_{G}(H))$;(3)the$\sigma$-Vasi{l}^{\mathrm{\prime }}ev-Murashkograph$\Gamma_{{\mathfrak{N}\sigma }}(G)$of$G$inwhich$(\sigma_i, \sigma_j)\in E(\Gamma{{\mathfrak{N}\sigma}}(G))$ifforsome${\mathfrak{N}{\sigma }}$-criticalsubgroup$H$of$G$wehave$\sigma_i \in \sigma (H)$and$\sigma_j\in \sigma (H/F_{{\sigma_i}}(H))$. In this paper, we study the structure of $G$ depending on the properties of these three graphs of $G$.