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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References
. Cayley, Arthur. "A Memoir on the Theory of Matrices." Philosophical Transactions of the Royal Society of London, vol. 148, 1858, pp. 17-37.
. Sylvester, James Joseph. "On the Principles of the Calculus of Forms." Cambridge and Dublin Mathematical Journal, vol. 1, 1846, pp. 52-97.
. Sylvester, James Joseph. "On the Theory of Matrices." Philosophical Magazine, vol. 5, 1851, pp. 428-436.
. Frobenius, Ferdinand Georg. "Über lineare Substitutionen und bilineare Formen." Journal für die reine und angewandte Mathematik, vol. 84, 1878, pp. 1-63.
. Eastman, B., Kim, I.-J., Shader, B.L., Vander Meulen, K.N. "Companion matrix patterns." Linear Algebra and its Applications, vol. 495, 2016, pp. 202-231.
. G. Chartrand, L. Lesniak, Graphs and Digraphs, third ed., Chapman & Hall, London, 1996.
. F. Harary, Graph Theory, Addison-Wesley Publ. Company, London, 1972.
Lipkovski, Aleksandar T. "Digraphs associated with finite rings." Publications de l'Institut Mathematique 92.106 (2012): 35-41.
. Daoub, Hamza, Osama Shafah, and Aleksandar Lipkovski. "An association between digraphs and rings." Filomat 36.3 (2022): 715-720.