Let us consider the finite monogenic semigroup with zero having elements . There exists an undirected graph associated with whose vertices are the non-zero elements and, f or , any two distinct vertices and are adjacent if .
In this paper, the diameter, girth, maximum and minimum degrees, domination number, chromatic number, clique number, degree sequence, irregularity index and also perfectness of 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 and , we present the spectral properties to the Cartesian product .
MSC: 05C10, 05C12, 06A07, 15A18, 15A36.