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Spectra of Boolean Graphs Over Finite Fields of Characteristic Two

Part of: Finite commutative rings Graph theory

Published online by Cambridge University Press:  04 November 2019

D. Scott Dillery  and
John D. LaGrange
D. Scott Dillery
Affiliation:
School of Mathematics and Sciences, Lindsey Wilson College, Columbia, KY 42728-1223, USA Email: [email protected]@lindsey.edu
John D. LaGrange
Affiliation:
School of Mathematics and Sciences, Lindsey Wilson College, Columbia, KY 42728-1223, USA Email: [email protected]@lindsey.edu
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Abstract

With entries of the adjacency matrix of a simple graph being regarded as elements of $\mathbb{F}_{2}$, it is proved that a finite commutative ring $R$ with $1\neq 0$ is a Boolean ring if and only if either $R\in \{\mathbb{F}_{2},\mathbb{F}_{2}\times \mathbb{F}_{2}\}$ or the eigenvalues (in the algebraic closure of $\mathbb{F}_{2}$) corresponding to the zero-divisor graph of $R$ are precisely the elements of $\mathbb{F}_{4}\setminus \{0\}$ . This is achieved by observing a way in which algebraic behavior in a Boolean ring is encoded within Pascal’s triangle so that computations can be carried out by appealing to classical results from number theory.

Keywords

zero-divisor graphBoolean ringeigenvaluePascal matrix

MSC classification

Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Secondary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 13M99: None of the above, but in this section
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 1 , March 2020 , pp. 58 - 65
Copyright
© Canadian Mathematical Society 2019

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