On a graph of monogenic semigroups

WOS: 000323562300004 ; 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 Gamma (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 xi and xj 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 Gamma (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 Gamma (S-M(1)) and Gamma (S-M(2)), we present the spectral properties to the Cartesian product Gamma (S-M(1)) square Gamma (S-M(2)). ; Faculty Research Fund, Sungkyunkwan University; Sungkyunkwan University BK21 Project, BK21 Math Modelling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea; Research Project Office of Selcuk UniversitySelcuk University ; The first author is supported by the Faculty Research Fund, Sungkyunkwan University, 2012 and Sungkyunkwan University BK21 Project, BK21 Math Modelling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea. The second and third authors are both partially supported by the Research Project Office of Selcuk University. Some of the material in this paper can also be found in the second author's Ph.D. thesis.