The total graph of a commutative semiring with respect to proper ideals

Let I be a proper ideal of a commutative semiring R and let P(I) be the set of all elements of R that are not prime to I. In this paper, we investigate the total graph of R with respect to I, denoted by T(\Gamma_{I} (R)). It is the (undirected) graph with elements of R as vertices, and for distinct x, y \in R, the vertices x and y are adjacent if and only if x + y \in P(I). The properties and possible structures of the two (induced) subgraphs P(\Gamma_{I} (R)) and \bar {P}(\Gamma_{I} (R)) of T(\Gamma_{I} (R)), with vertices P(I) and R - P(I), respectively are studied.

Keywords:

Commutative semirings , Zero-divisor , Total graph