Directed Graphs of Frobenius Companion Matrices of Order 2
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Abstract
In this work, we explore the set of Frobenius companion matrices with entries from the ring of integers modulo a prime $p$, denoted as $M_2^{(F)}(\mathbb{Z}_p)$. We introduce a mapping $\psi$ to construct a directed graph $G_p$, where vertices represent these matrices and edges are defined by $\psi$. The study focuses on the structure and properties of $G_p$, including vertex degrees, cycles, and connected components, in relation to the eigenvalues of the matrices.
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