On a graph of monogenic semigroups
Let us consider the finite monogenic semigroup S M with zero having elements { x , x 2 , x 3 , … , x n } . There exists an undirected graph Γ ( S M ) associated with S M whose vertices are the non-zero elements x , x 2 , x 3 , … , x n and, f or 1 ≤ i , j ≤ n , any two distinct vertices x i and x j a...
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Veröffentlicht in: | Journal of inequalities and applications 2013-12, Vol.2013 (1), p.1-13, Article 44 |
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Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
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Zusammenfassung: | Let us consider the finite monogenic semigroup
S
M
with zero having elements
{
x
,
x
2
,
x
3
,
…
,
x
n
}
. There exists an undirected graph
Γ
(
S
M
)
associated with
S
M
whose vertices are the non-zero elements
x
,
x
2
,
x
3
,
…
,
x
n
and,
f
or
1
≤
i
,
j
≤
n
, any two distinct vertices
x
i
and
x
j
are adjacent if
i
+
j
>
n
.
In this paper, the diameter, girth, maximum and minimum degrees, domination number, chromatic number, clique number, degree sequence, irregularity index and also perfectness of
Γ
(
S
M
)
have been established. In fact, some of the results obtained in this section are sharper and stricter than the results presented in DeMeyer
et al.
(Semigroup Forum 65:206-214, 2002). Moreover, the number of triangles for this special graph has been calculated. In the final part of the paper, by considering two (not necessarily different) graphs
Γ
(
S
M
1
)
and
Γ
(
S
M
2
)
, we present the spectral properties to the Cartesian product
Γ
(
S
M
1
)
□
Γ
(
S
M
2
)
.
MSC:
05C10, 05C12, 06A07, 15A18, 15A36. |
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ISSN: | 1029-242X 1025-5834 1029-242X |
DOI: | 10.1186/1029-242X-2013-44 |