On Conjugacy Class Graph of Normal Subgroup

Abstract

Let [Formula: see text] be a finite group and [Formula: see text] a normal subgroup of [Formula: see text]. Denote by [Formula: see text] the graph whose vertices are all distinct [Formula: see text]-conjugacy class sizes of non-central elements in [Formula: see text], and two vertices of [Formula: see text] are adjacent if and only if they are not coprime numbers. We prove that if the center [Formula: see text] and [Formula: see text]is [Formula: see text]-regular for [Formula: see text], then either a section of [Formula: see text]is a quasi-Frobenius group or [Formula: see text] is a complete graph with [Formula: see text] vertices.