EIGENVALUES OF BOOLEAN GRAPHS AND PASCAL-TYPE MATRICES

Year 2013, Volume: 13 Issue: 13, 109 - 119, 01.06.2013

John D. Lagrange

Abstract

Let R be a commutative ring with 1 6= 0. The zero-divisor graph of
R is the (undirected) graph whose vertices consist of the nonzero zero-divisors
of R such that two distinct vertices x and y are adjacent if and only if xy = 0.
Given an integer k > 1, let Ak be the adjacency matrix of the zero-divisor
graph of the finite Boolean ring of order 2k. In this paper, it is proved that
the eigenvalues of Ak are completely determined by the eigenvalues given by
two (k − 1) × (k − 1) Pascal-type matrices Pk and Qk. Multiplicities are also
determined.

Keywords

zero-divisor graph, adjacency matrix, Boolean ring

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• Division of Natural and Behavioral Sciences
• Lindsey Wilson College
• Columbia, Kentucky 42728, USA
• e-mail: [e-mail]