Generalization of the non-commuting graph of a group via a normal subgroup

Fereshteh Kakeri^a, Ahmad Erfanian^b,*, Farzaneh Mansoori^a

ABSTRACT: Let G be a finite group and N be a normal subgroup of G. We define an undirected simple graph Γ_N,G to be a graph whose vertex set is all elements in G∖Z^N(G) and two vertices x and y are adjacent iff [x,y]∉N, where Z^N(G)={g∈G:[x,g]∈N for all x∈G}. If N=1, then we obtain the known non-commuting graph of G. We give some basic results about connectivity, regularity, planarity, 1-planarity and some numerical invariants of the graph which are mostly improvements of the results given for non-commuting graphs. Also, a probability related to this graph is defined and a formula for the number of edges of the graph in terms of this probability is given.

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^a	Ferdowsi University of Mashhad, International Campus, Mashhad, Iran
^b	Department of Pure Mathematics and Centre of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, Mashhad, Iran

* Corresponding author, E-mail: erfanian@math.um.ac.ir

Received 18 Feb 2015, Accepted 0 0000

Generalization of the non-commuting graph of a group via a normal subgroup

Fereshteh Kakeria, Ahmad Erfanianb,*, Farzaneh Mansooria

Fereshteh Kakeri^a, Ahmad Erfanian^b,*, Farzaneh Mansoori^a