Spectra of Boolean Graphs Over Finite Fields of Characteristic Two

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 alg...

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Published in:	Canadian mathematical bulletin Vol. 63; no. 1; pp. 58 - 65
Main Authors:	Dillery, D. Scott, LaGrange, John D.
Format:	Journal Article
Language:	English
Published:	Montreal Cambridge University Press 01.03.2020
Subjects:	Algebra Boolean Boolean algebra Commutativity Eigenvalues Fields (mathematics) Mathematics Number theory Rings (mathematics)
ISSN:	0008-4395, 1496-4287
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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.
AbstractList	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. 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.
Author	Dillery, D. Scott LaGrange, John D.
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Snippet	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... 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...
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SubjectTerms	Algebra Boolean Boolean algebra Commutativity Eigenvalues Fields (mathematics) Mathematics Number theory Rings (mathematics)
Title	Spectra of Boolean Graphs Over Finite Fields of Characteristic Two
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