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
Hauptverfasser: Das, K Ch, Akgüneş, Nihat, Çevik, A Sinan
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Sprache:eng
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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.
ISSN:1029-242X
1025-5834
1029-242X
DOI:10.1186/1029-242X-2013-44