(untitled) (European Journal of Pure and Applied Mathematics, 2025)
A unitary Cayley graph Γ_n of a finite cyclic ring Z_n is a graph with vertex set Z_n and two vertices x and y are adjacent if and only if x-y is a unit in Z_n or equivalently, (x-y,n)=1. A nonempty subset A of Z_n is called an acyclic set of Γ_n if a subgraph of Γ_n induced by A contains no cycles. The maximum cardinality among the acyclic sets of Γ_n is called the upper acyclic number of Γ_n and is denoted by Λ(Γ_n). Moreover, the maximum number k of vertices of Γ_n in which every subgraph of Γ_n induced by k vertices contains no cycles is called the lower acyclic number of Γ_n and denoted by λ(Γ_n). In this paper, we determine the lower and upper acyclic numbers for unitary Cayley graphs of Z_n and their complements.