Normal subgroup based power graph of finite groups

Abstract

Let G be a finite group and $N \unlhd G$. The normal subgroup based power graph of G, $\Gamma _N(G)$, is an undirected graph with vertex set $(G{\setminus }N) \cup \{e\}$ in which two distinct vertices a and b are adjacent if and only if $aN = b^mN$ or $bN = a^nN$, for some positive integers m and n. The aim of this paper is to characterize all pairs (G, N) of a finite group G and a proper non-trivial normal subgroup N of G such that $\Gamma _{H}(G)$ is a split, bisplit and $(n-1)-$bisplit. A classification of finite groups with unicyclic and tricyclic normal subgroup based power graph are also given and it is proved that there is no group containing a proper and non-trivial normal subgroup such that its normal subgroup based power graph is a bicyclic or tetracyclic graph.