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2020•Journal of Zankoy Sulaimani - Part AOpen access

Ideal Graphs Supported By Given Ideals of Commutative Rings

F. H. Abdulqadr

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Abstract

In this paper we introduce and study a new kind of graph that constructed by non-trivial ideals of a commutative ring with identity. Let R be a commutative ring with identity and P be a non-trivial ideal of R. The ideal graph supported by the ideal P, denoted by (P), is the undirected graph whose vertices are those non-trivial ideals I of R such that there exists a non-trivial ideal JI of R with IJ⊂P, and every two vertices I and J are adjacent if IJ and IJ⊂P. We investigate the connectivity, completeness and planarity of the graph (P). Also we explore the diameter, girth, domination, clique number and chromatic number of (P).

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What this paper is about

In this paper we introduce and study a new kind of graph that constructed by non-trivial ideals of a commutative ring with identity. Let R be a commutative ring with identity and P be a non-trivial ideal of R. The ideal graph supported by the ideal P, denoted by (P), is the undirected graph whose vertices are those non-trivial ideals I of R such that there exists a non-trivial ideal JI of R with IJ⊂P, and every two vertices I and J are adjacent if IJ and IJ⊂P. We investigate the connectivity, completeness and planarity of the graph (P). Also we explore the diameter, girth, domination, clique number and chromatic number of (P).

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Available abstract

In this paper we introduce and study a new kind of graph that constructed by non-trivial ideals of a commutative ring with identity. Let R be a commutative ring with identity and P be a non-trivial ideal of R. The ideal graph supported by the ideal P, denoted by (P), is the undirected graph whose vertices are those non-trivial ideals I of R such that there exists a non-trivial ideal JI of R with IJ⊂P, and every two vertices I and J are adjacent if IJ and IJ⊂P. We investigate the connectivity, completeness and planarity of the graph (P). Also we explore the diameter, girth, domination, clique number and chromatic number of (P).

Key concepts: Clique number, Mathematics, Commutative ring, Combinatorics, Ideal (ethics), Radical of an ideal, Planarity testing, Discrete mathematics

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