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Spectra of Boolean Graphs Over Finite Fields of Characteristic Two
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 63 / Issue 1 / March 2020
- Published online by Cambridge University Press:
- 04 November 2019, pp. 58-65
- Print publication:
- March 2020
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- Article
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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.
