In a previous paper, the author proved that non-eigenvalues of the adjacency operator of an infinite locally finite connected graph over a field of characteristic 0 can be only algebraic over the prime subfield of the field elements (in particular, only algebraic numbers when the field is $\mathbb{C}$). There were also given examples of infinite locally finite connected graphs for which certain algebraic numbers are not eigenvalues of their adjacency operators over $\mathbb{C}$. In the present paper we give examples of infinite locally finite connected graphs for each of which infiniely many algebraic numbers are not eigenvalues of its adjacency operator over $\mathbb{C}$. More exactly, for every prime integer $p$, we construct an infinite locally finite connected graph such that no positive integer multiple of $p$ is an eigenvalue of the adjacency operator over $\mathbb{C}$ of the graph. In addition, in the paper a necessary condition (based on results of the mentioned previous paper) is given for an algebraic number not to be an eigenvalue of the adjacency operator over $\mathbb{C}$ of at least one infinite locally finite connected graph.
Keywords: locally finite graph, adjacency matrix, eigenvalue
Received November 7, 2024
Revised November 14, 2024
Accepted November 18, 2024
Funding Agency: The work was supported under state contract of IMM UB RAS, project no. FUMF-2022-0003.
Vladimir Ivanovich Trofimov, Dr. Phys.-Math. Sci., Lead. Sci. Researcher, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Professor, Ural Federal University, Yekaterinburg, 620083 Russia,
e-mail: trofimov@imm.uran.ru
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Cite this article as: V.I. Trofimov. Infinite locally finite connected graphs with countable complements in $\mathbb{C}$ of the sets of eigenvalues, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 1, pp. 228–235.
