Search

The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. Let R be an associative (not necessarily commutative) ring. We focus on the domination number of the zero-divisor graph Γ (R) , the compressed zero-divisor graph Γ E (R) and the unit graph G (R). We find some relations between the domination number of the zero-divisor graph and that of the compressed zero-divisor graph. Moreover, some relations between the domination number of Γ (R) and Γ (R [ x ; α , δ ]) , as well as the relations between the domination number of G (R) and G (R [ [ x ; α ] ]) , are studied. [ABSTRACT FROM AUTHOR]

For a finite commutative ring R, the square mapping graph of R is a directed graph Γ(R) whose set of vertices is all the elements of R and for which there is a directed edge from a to b if and only if a2=b. We establish necessary and sufficient conditions for the existence of isolated fixed points, and the cycles with length greater than 1 in Γ(R). We also examine when the induced subgraph on the set of zero-divisors of a local ring with odd characteristic is semiregular. Moreover, we completely determine the finite commutative rings whose square mapping graphs have exactly two, three or four components. [ABSTRACT FROM AUTHOR]