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Reciprocal eigenvalue properties using the zeta and Möbius functions.
- Source :
-
Linear Algebra & its Applications . Aug2024, Vol. 694, p186-205. 20p. - Publication Year :
- 2024
-
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 B n of order 2 n. 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 B n. 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00243795
- Volume :
- 694
- Database :
- Academic Search Index
- Journal :
- Linear Algebra & its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 177289639
- Full Text :
- https://doi.org/10.1016/j.laa.2024.04.010