Total graph of a commutative semiring with respect to singular ideal
DOI:
https://doi.org/10.22199/issn.0717-6279-2020-03-0032Keywords:
Semiring, Total graph, Singular ideal, Induced subgraphAbstract
Let S be a commutative semiring with unity. The singular ideal Z(S) of S is defined as Z(S) = {s ∈ S | sK = 0 for some essential ideal K of S}. In this paper, we introduce the notion of total graph of a commutative semiring with respect to the singular ideal. We define this graph as the undirected graph T(Γ(S)) with all elements of S as vertices and any two distinct vertices x and y are adjacent if and only if x + y ∈ Z(S). We discuss various characteristics of this total graph and also characterize some important properties of certain induced subgraphs of this total graph.
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