On the regular graph of ideals of total ring of fractions

The regular graph of ideals of a commutative ring R, denoted by Gamma(reg)(R), is a graph whose vertex-set is the set of all nontrivial ideals of R and, for every two distinct vertices I and J, there is an edge between I and J, whenever I contains a nonzero zero-divisor on J or vice versa. In this paper, we will show that Gamma(reg)(T(R)) is isomorphic to an induced sub-graph of Gamma(reg)(R) which nicely can describe Gamma(reg)(R). This approach enables us to find the independence number of Gamma(reg)(R) when R is a reduced ring. Among other things, some basic graph theoretic properties of Gamma(reg)(T(R)) and relevant ring theoretic properties of R will be studied.