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Let R be an I-semiring and S(R) be the set of all identity-summand elements of R. In this paper we introduce the total graphof R with respect to identity-summand elements, denoted by T(Γ(R)),and investigate basic properties of S(R) which help us to gain interestingresults about T(Γ(R)) and its subgraphs. 1. IntroductionAssociating a graph to an algebraic structure is a research subject and hasattracted considerable attention. In fact, the research in this subject aims atexposing the relationship between algebra and graph theory and at advancingthe application of one to the other.In 1988, Beck [11] introduced the idea of a zero-divisor graph of a commu-tative ring R with identity. This notion was later redefined by Anderson andLivingston in [6]. Since then, there has been a lot of interest in this subject andvarious papers were published establishing different properties of these graphsas well as relations between graphs of various extensions. The total graph ofa commutative ring was introduced by Anderson and Badawi in [3], as thegraph with all elements of R as vertices, and two distinct vertices x,y∈ Rareadjacent if and only if x+ y∈ Z(R) where Z(R) is the set of all zero divisorof R. In [4], Anderson and Badawi studied the subgraph T |
