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

Authors : G KALAIMURUGAN, P VIGNESH, M AFKHAMI, Z BARATI

Pages : 18-33

Doi:10.24330/ieja.1152714

View : 15 | Download : 5

Publication Date : 2023-01-09

Article Type : Research Paper

Abstract :Let $R$ be a commutative ring without identity. The zero-divisor graph of $R,$ denoted by $\\Gammainsert ignore into journalissuearticles values(R);$ is a graph with vertex set $Zinsert ignore into journalissuearticles values(R);\\setminus \\{0\\}$ 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