Planar index and outerplanar index of zero-divisor graphs of commutative rings without identity

Year 2023, , 18 - 33, 09.01.2023

G. Kalaımurugan P. Vıgnesh M. Afkhamı Z. Baratı

https://doi.org/10.24330/ieja.1152714

Abstract

Let $R$ be a commutative ring without identity. The zero-divisor graph of $R,$ denoted by $\Gamma(R)$ is a graph with vertex set $Z(R)\setminus \{0\}$ which is the set of all nonzero zero-divisor elements of $R,$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0.$ In this paper, we characterize the rings whose zero-divisor graphs are ring graphs and outerplanar graphs. Further, we establish the planar index, ring index and outerplanar index of the zero-divisor graphs of finite commutative rings without identity.

Keywords

Zero-divisor graph, line graph, iterated line graph, planar index, outerplanar index, ring index

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