This note shows the limitations of n-Cayley graphs over solvable groups in forming expander families, suggesting important structural insights.
A graph Γ is called an n-Cayley graph over a group G if there exists a semiregular subgroup of Aut(Γ) that is isomorphic to G with n orbits (of equal size). This is one of the generalizations of Cayley graphs. In this paper, we show that a sequence of n-Cayley graphs over solvable groups cannot form an expander family. We also show that a connected graph being an n-Cayley graph over G is equivalent to being a Galois covering with Galois group G.
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Nao Toyama (2024) studied this question.
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