Reciprocal eigenvalue properties using the zeta and Möbius functions

Abstract

In this paper, we develop a new approach to study the spectral properties of Boolean graphs using the zeta and Möbius functions on the Boolean algebra Bn of order 2n. This approach yields new proofs of the previously known results about the reciprocal eigenvalue property of Boolean graphs. Further, this approach allows us to extend the results to a more general setting of the zero-divisor graphs Γ(P) of complement-closed and convex subposets P of Bn. To do this, we consider the left linear representation of the incidence algebra of a poset P on the vector space of all real-valued functions V(P) on P. We then write down the adjacency operator A of the graph Γ(P) as the composition of two linear operators on V(P), namely, the operator that multiplies elements of V(P) on the left by the zeta function ζ of P and the complementation operator. This allows us to obtain the determinant of A and the inverse of A in terms of the Möbius function μ of the complement-closed posets P. Additionally, if we impose convexity on the poset P, then we obtain the strong reciprocal or strong anti-reciprocal eigenvalue property of Γ(P) and also obtain the absolute palindromicity of the characteristic polynomial of A. This produces a large family of examples of graphs having the strong reciprocal or strong anti-reciprocal eigenvalue property.