Boolean rings and reciprocal eigenvalue properties

Let A be the adjacency matrix of the zero-divisor graph Γ(R) of a finite commutative ring R containing nonzero zero-divisors. In this paper, it is shown that Γ(R) is the zero-divisor graph of a Boolean ring if and only if det(A)=-1. Also, A is similar to plus or minus its inverse whenever R is a Boolean ring. As a consequence, it is proved that Γ(R) is the zero-divisor graph of a Boolean ring if and only if the set of eigenvalues (including multiplicities) of Γ(R) can be partitioned into 2-element subsets of the form {λ,±1/λ}. Furthermore, any finite Boolean ring R is characterized by the degree and coefficients of the characteristic polynomial of A.