Let Rn be a finite reduced ring with n maximal ideals 픪i and let Γ(Rn) be the zero-divisor graph associated to Rn. The class of rings Rn contains the Boolean rings as a subclass. When Rn/픪i = 픽 for all i where 픽 is a finite field, we associate two (n − 1) × (n − 1) sized matrices P and Q to the graph Γ(Rn) having combinatorial entries and use these matrices to determine the spectrum of this graph. More precisely, we show that every eigenvalue of P and of − Q is an eigenvalue of Γ(Rn). To do this, we give a recursive description of the adjacency matrix of this graph and also exhibit its equitable partition. This is used in computing the determinant, rank and nullity of the adjacency matrix. Further, we propose that the eigenvalues of P, − Q and the eigenvalue 0 exhaust all the eigenvalues of Γ(Rn). [ABSTRACT FROM AUTHOR]