Lower and Upper Acyclicities on Unitary Cayley Graphs of Finite Commutative Rings (European Journal of Pure and Applied Mathematics, 2025)
A unitary Cayley graph $Γ_n$ of a finite cyclic ring $\mathbb{Z}_n$ is a graph with vertex set $\mathbb{Z}_n$ and two vertices $x$ and $y$ are adjacent if and only if $x-y$ is a unit in $\mathbb{Z}_n$ or equivalently, $\gcd(x-y,n)=1$. A nonempty subset $A$ of $\mathbb{Z}_n$ 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 $\mathbb{Z}_{n}$ and their complements.
Keywords: Acyclic sets, Lower acyclic numbers, Upper acyclic numbers, Unitary Cayley graphs