On graphs with generalized reciprocal eigenvalue property

We introduce the notion of strong generalized reciprocal eigenvalue property (SGR) for graphs by extending the notion of strong reciprocal eigenvalue property (SR). A graph G is said to possess property (SGR) if there exists positive integer m such that for every non-zero eigenvalue λ of G occurring with multiplicity k, m/λ is an eigenvalue of G occurring with the same multiplicity k. When m is a negative integer, we call this the strong generalized anti-reciprocal eigenvalue property (SGAR). We show that the zero-divisor graphs of the finite direct product of chains satisfy the property (SGR) or property (SGAR). This generalizes the result of LaGrange, which states that the Boolean graph satisfies the strong reciprocal eigenvalue property. As a consequence, it follows that the zero-divisor graphs associated to finite reduced rings have the generalized reciprocal eigenvalue property.