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ScienceAsia 42(2016): 231-235 |doi: 10.2306/scienceasia1513-1874.2016.42.231

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

Fereshteh Kakeria, Ahmad Erfanianb,*, Farzaneh Mansooria

ABSTRACT:     Let G be a finite group and N be a normal subgroup of G. We define an undirected simple graph ΓN,G to be a graph whose vertex set is all elements in GZN(G) and two vertices x and y are adjacent iff [x,y]∉N, where ZN(G)={gG:[x,g]∈N for all xG}. 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