Let G be a metacyclic p-group and Z(G) be its center. The non-commuting graph ΓG of a metacyclic p-group G is defined as the graph whose vertex set is G-Z(G) and two distinct vertices x and y are connected by an edge if and only if [x, y] ≠ 1. It is proved that the clique number and chromatic number of graph ΓG associated with the metacyclic p-group G are identical.
REFERENCES
1.
A.
Abdollahi
, S.
Akbari
, H. R.
Maimani
, Non-commuting graph of a group
, J. Algebra
298
, 468
−492
(2006
).2.
I.
Beck
, Coloring of commutative rings
, J. Algebra
116
, 208
−226
(1988
).3.
E. A.
Bertram
, Some applications of graph theory to finite groups
, Discrete Math.
44
, 31
−43
(1983
).4.
J. R.
Beuerle
, An elementary classification of finite metacyclic p-groups of class at least three
, Algebra Colloq.
21
, 467
−472
(2005
).5.
P.
Erdos
, and P.
Turan
, On some problems of statistical group theory
, Acta Math. Acad. of Sci. Hung.
19
413
−435
(1968
).6.
7.
M. R.
Darafsheh
, Groups with the same noncommuting graph
, Discrete Appl. Math.
157
, 833
−837
(2009
).8.
A.
Erfanian
, R.
Rezaei
, and P.
Lescot
, On the relative commutativity degree of a subgroup of a finite group
, Comm. in Algebra
35
, 4183
−4197
(2007
).9.
W. H.
Gustafson
, What is the probability that two groups elements commute?
Amer. Math. Monthly
80
, 1031
−1304
(1973
).10.
A. R.
Moghaddamfar
, W. J.
Shi
, W.
Zhou
, A. R.
Zokayi
, On non-commuting graph associated with a finite group
, Siberian Math. J.
46
, 325
−332
(2005
).11.
K.
Moradipour
, N. H.
Sarmin
, A.
Erfanian
, Conjugacy classes and commuting probability in finite metacyclic p-groups
, Science Asia
38
(1
), 113
−117
(2012
).12.
K.
Moradipour
, N. H.
Sarmin
, A.
Erfanian
, Conjugacy classes and commutativity degree of metacyclic 2-groups
, Comptes Rendus de L’Academie Bulgare des Sciences
66
(10
), 1363
–1372
(2013
).13.
B. H.
Neumann
, A problem of Paul Erdos on groups
, J. Aust. Math. Soc. Ser. A
21
, 467
−472
(1976
).14.
Y.
Segev
, The commuting graph of minimal nonsolvable groups
, Geom. Dedicata
88
(13
), 55
−66
(2001
).
This content is only available via PDF.
© 2017 Author(s).
2017
Author(s)
You do not currently have access to this content.