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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| Veröffentlicht in: | Canadian mathematical bulletin Jg. 63; H. 1; S. 58 - 65 |
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| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Montreal
Cambridge University Press
01.03.2020
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| Schlagworte: | |
| ISSN: | 0008-4395, 1496-4287 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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. |
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| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0008-4395 1496-4287 |
| DOI: | 10.4153/S0008439519000365 |