$\pi(G)$ denote the set of prime divisors of the order of $G$ and denote by $Z(G)$ the center of $G$. The\textit{ prime graph} of $G$ is the graph with vertex set $\pi(G)$ where two distinct primes $p$ and $q$ are joined by an edge if and only if $G$ contains an element of order $pq$ and the \textit{non-commuting graph} of $G$ is the graph with the vertex set $G-Z(G)$ where two non-central elements $x$ and $y$ are joined by an edge if and only if $xy \neq yx$.
Let $ G $ and $ H $ be non-abelian finite groups with isomorphic non-commuting graphs. In this article, we show that if $ | Z ( G ) | = | Z ( H ) | $, then $ G $ and $ H $ have the same prime graphs and also, the set of orders of the maximal abelian subgroups of $ G $ and $ H $ are the same.

A. Abdollahi, S. Akbari and H.R. Maimani (2006). Non-commuting graph of a group. J. Algebra. 298 (2), 468-492

2

N. Ahanjideh and A. Iranmanesh On the orders of maximal abelian subgroups of
Bn(q).

3

G.Y. Chen (2006). A characterization of alternating groups by the set of orders of their maximal abelian subgroups. Siberian Math. J.. 47 (3), 594-596

4

M.R. Darafsheh (2009). Groups with the same non-commuting graph. Discrete Appl. Math.. 157 (4), 833-837

5

A. Iranmanesh and A. Jafarzadeh (2008). On the commuting graph associated with the symmetric and alternating groups. J. Algebra Appl.. 7 (1), 129-146

6

E.I. Khukhro and V.D. Mazurov (Editors) (2010). Unsolved problems in group theory: The Kourovka Notebook. Sobolev Institute of Mathematics, Novosibirsk. 17th edition

7

A.S. Kondratev (1990). Prime graph components of finite simple groups. Math. USSR-Sb.. 67 (1), 235-247

8

A.V. Vasil'ev (2005). On a relation between the structure of a
finite group and the properties of its prime graph. Siberian Math. J.. 46 (3), 396-404

9

A.V. Vasil'ev and E. P. Vdovin (2005). An adjacency criterion in the prime graph of a finite simple group. Algebra Logic. 44 (6), 381-406

10

J.S. Williams (1981). Prime graph components of finite groups. J. Algebra. 69 (2), 487-513