Abstract
Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in are adjacent if and only if and . In this paper, we investigate some combinatorial properties of the cozero-divisor graphs and such as connectivity, diameter, girth, clique numbers and planarity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.
Highlights
Let W R be the set of all non-unit elements of R.For an arbitrary commutative ring R, the cozero-divisor graph of R, denoted by R, was introduced in [6], which is a dual of zero-divisor graph R “in some sense”
We study the cozero-divisor graphs of the direct products of two arbitrary commutative rings
We study the cozero-divisor graphs of the rings of polynomials, power series and the direct product of two arbitrary commutative rings
Summary
Let W R be the set of all non-unit elements of R. For an arbitrary commutative ring R, the cozero-divisor graph of R, denoted by R , was introduced in [6], which is a dual of zero-divisor graph R “in some sense”. Two distinct vertices a and b in W R , a is adjacent to b if and only if a bR and b aR , where cR is an ideal generated by the element c in R. We study the cozero-divisor graphs of the rings of polynomials, power series and the direct product of two arbitrary commutative rings. In a graph G, the distance between two distinct vertices a and b, denoted by dG a,b , is the length of the shortest path connecting a and b, if such a path exists; otherwise, we set dG a,b :. The diameter of a graph G is diam G sup dG a,b a and b are distinct vertices of G
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