Normal Subgroup Based Power Graph of a finite Group

For a finite group $G$ with a normal subgroup $H$, the normal subgroup based power graph of $G$, denoted by $Γ_H(G)$ whose vertex set $V(Γ_H(G))=(G\setminus H)\bigcup \{e\}$ and two vertices $a$ and $b$ are edge connected if $aH=b^mH$ or $bH=a^nH$ for some $m, n \in \mathbb{N}$. In this paper we obtain some fundamental characterizations of the normal subgroup based power graph. We show some relation between the graph $Γ_H(G)$ and the power graph $Γ(\frac{G}{H})$. We show that $Γ_H(G)$ is complete if and only of $\frac{G}{H}$ is cyclic group of order $1$ or $p^m$, where $p$ is prime number and $m\in \mathbb{N}$. $Γ_H(G)$ is planar if and only if $|H|=2$ or $3$ and $\frac{G}{H}\cong \mathbb{Z}_2\times \mathbb{Z}_2 \times \cdots \times \mathbb{Z}_2$. Also $Γ_H(G)$ is Eulerian if and only if $|G|\equiv |H|$ mod$2$.

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