Spectra of Boolean Graphs Over Finite Fields of Characteristic Two

Dillery, D. Scott ; LaGrange, John D.

Canadian mathematical bulletin, Tome 63 (2020) no. 1, pp. 58-65

Résumé

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.

Détail

DOI : 10.4153/S0008439519000365

Mots-clés : zero-divisor graph, Boolean ring, eigenvalue, Pascal matrix