Order Conjecture for non-commuting graph of a group

The following was conjectured by A. Abdollahi, S. Akbari and H. R. Maimani in (Non-commuting graph of a group, Journal of Algebra, 298 (2006) 468-492.)

Order Conjecture. If $G$ and $H$ are two non-abelian finite groups with isomorphic non-commuting graphs, then $|G|=|H|$.

It was proved that Order Conjecture is true if and only if it is true for all non-abelian solvable finite $AC$-groups. By an $AC$-group, we mean a group in which the centralizer of every non-central element is abelian.

The order Conjecture has been refuted in the following paper

[*] A. R. Moghaddamfar, On non-commutating graphs, Siberian Math. J. 47 (2006), no. 5, 911-914.

It is mentioned in [*] that the example given in the article is due to M. Isaacs.

Title	Order Conjecture for non-commuting graph of a group
Canonical name	OrderConjectureForNoncommutingGraphOfAGroup
Date of creation	2013-03-22 15:18:53
Last modified on	2013-03-22 15:18:53
Owner	abdollahi (9611)
Last modified by	abdollahi (9611)
Numerical id	10
Author	abdollahi (9611)
Entry type	Conjecture
Classification	msc 20D60