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(􀀀I (R)). It is the (undirected) graph with elements of R as vertices, and for distinct x; y 2 R, the vertices x and y are adjacent if and only if x + y 2 P(I). The properties and possible structures of the two (induced) subgraphs P(􀀀I (R)) and  P(􀀀I (R)) of T(􀀀I (R)), with vertices P(I) and R 􀀀 P(I), respectively are studied.