Directed Graphs of Frobenius Companion Matrices of Order 2

In this work, we investigate Frobenius companion matrices with entries from the ring of integers modulo a prime number. We introduce a mapping 𝝍 to construct a directed graph, where vertices represent these matrices and edges are defined by 𝝍. Our analysis focuses on the structure and properties of this directed graph, including vertex degrees,cycles, and connected components, in relation to the eigenvalues of the matrices،