Maximal ideal graph of commutative semirings

In this paper a new kind of graph on a commutative semiring is introduced and investigated. The maximal ideal graph of S, denoted by MG(S), is a graph with all nontrivial ideals of S as vertices and two distinct vertices I and J are adjacent if and only if I + J is a maximal ideal of S. In this article, some interrelation between the graph theoretic properties of this graph and some algebraic properties of semirings are studied. We investigated the basic properties of the maximal ideal graph such as diameter, girth, clique number, cut vertex, planar property.