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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| Vydáno v: | Canadian mathematical bulletin Ročník 63; číslo 1; s. 58 - 65 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Montreal
Cambridge University Press
01.03.2020
|
| Témata: | |
| 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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| Cites_doi | 10.1080/00927870701509156 10.1016/j.laa.2017.04.017 10.2307/2304500 10.1006/jabr.1998.7840 10.1007/978-1-4613-0163-9 10.1080/03081080600792897 10.1016/j.laa.2013.02.009 10.1016/j.laa.2011.05.042 |
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| References | S0008439519000365_r1 Atiyah (S0008439519000365_r2) 1969 Carlitz (S0008439519000365_r4) 1965; 3 S0008439519000365_r9 S0008439519000365_r7 Cvetković (S0008439519000365_r5) 1979 S0008439519000365_r11 S0008439519000365_r8 S0008439519000365_r12 S0008439519000365_r6 LaGrange (S0008439519000365_r10) 2013; 13 S0008439519000365_r3 |
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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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