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Isomorphisms and Planarity of Zero-Divisor Graphs

  • Jesse Gerald SmithJr.Email author
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

Let R be a commutative ring with nonzero identity and I a proper ideal of R. The zero-divisor graph of R, denoted by \(\varGamma (R)\), is the graph on vertices \(R^*=R\setminus \{0\}\) where distinct vertices x and y are adjacent if and only if \(xy=0\). The ideal-based zero-divisor graph of R with respect to the ideal I, denoted by \(\varGamma _I(R)\), is the graph on vertices \(\{x \in R\setminus I \mid xy\in I\) for some \(y\in R\setminus I \}\), where distinct vertices x and y are adjacent if and only if \(xy\in I\). In this paper, we cover two main topics: isomorphisms and planarity of zero-divisor graphs. For each topic, we begin with a brief overview on past research on zero-divisor graphs. Whereafter, we provide extensions of that material to ideal-based zero-divisor graphs.

Notes

Acknowledgements

I would like to thank my advisor, David F. Anderson, for his contribution and comments in my graduate research. This material is derived from dissertation research performed at the University of Tennessee, Knoxville [18]. Some of my fondest memories are discussing the rings, graphs, and isomorphisms found in this paper with this amazing man. I affectionately remember being called a trouble maker when meeting with him about the error that gave birth to my work on graph isomorphisms. We spent many hours talking about many things (sometimes things other than mathematics). David, I offer you my sincerest thanks. I would also like to thank my colleagues and mentors at Maryville College for encouraging me in the transition to a faculty member. Additional thanks to Chase Worley for helping me with edits to this document.

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© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Maryville CollegeMaryvilleUSA

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