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

Abstract

ABSTRACT A combinatorial approach is given to compute bases for eigenspaces of zero-divisor graphs of finite Boolean rings. A commutative monoid of graphs is shown to contain a cyclic submonoid that determines values of the entries of basis elements, while the members of its complement encode the supports of these elements. Furthermore, every member of 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.