Eigenvalues of zero-divisor graphs, Catalan numbers, and a decomposition of eigenspaces

ABSTRACT

A combinatorial approach is given to compute bases for eigenspaces of zero-divisor graphs of finite Boolean rings. A commutative monoid $\mathcal{S}$ of graphs is shown to contain a cyclic submonoid $\mathcal{U}$ that determines values of the entries of basis elements, while the members of its complement $\mathcal{S}\setminus \mathcal{U}$ encode the supports of these elements. Furthermore, every member of $\mathcal{S}\setminus \mathcal{U}$ is associated with a Catalan-triangle number, which counts the number of basis elements whose supports are determined by the given member. This is established by using a combinatorial interpretation of Catalan-triangle numbers to produce linearly independent sets of eigenvectors.

KEYWORDS:

• Zero-divisor graph
• adjacency matrix
• Boolean ring

MATHEMATICS SUBJECT CLASSIFICATION (2010)::

• 05C25
• 05C50
• 06E99

Disclosure statement

No potential conflict of interest was reported by the author(s).