Abstract
Let us consider groups
G
1
=
Z
k
∗
(
Z
m
∗
Z
n
)
,
G
2
=
Z
k
×
(
Z
m
∗
Z
n
)
,
G
3
=
Z
k
∗
(
Z
m
×
Z
n
)
,
G
4
=
(
Z
k
∗
Z
l
)
∗
(
Z
m
∗
Z
n
)
and
G
5
=
(
Z
k
∗
Z
l
)
×
(
Z
m
∗
Z
n
)
, where
k
,
l
,
m
,
n
≥
2
. In this paper, by defining a new graph
Γ
(
G
i
)
based on the Gröbner-Shirshov bases over groups
G
i
, where
1
≤
i
≤
5
, we calculate the diameter, maximum and minimum degrees, girth, chromatic number, clique number, domination number, degree sequence and irregularity index of
Γ
(
G
i
)
. Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in such fields as chemistry (in the meaning of atoms, molecules, energy
etc.
) and engineering (in the meaning of signal processing
etc.
), game theory and physics. In addition, the Gröbner-Shirshov basis and the presentations of algebraic structures contain a mixture of algebra, topology and geometry within the purposes of this journal.
MSC:
05C25, 13P10, 20M05, 20E06, 26C10.