Given a finite group G, we say that a subset C of G is power-closed if, for every x∈C and y∈⟨x⟩ with ⟨x⟩=⟨y⟩, we have y∈C. In this paper, we are interested in finite Cayley digraphs Cay(G,C) over G with connection set C, where C is a union of conjugacy classes of G. We show that each eigenvalue of Cay(G,C) is integral if and only if C is power-closed. This result will follow from a more general result.
Godsil, C., Spiga, P. (2025). Integral normal Cayley graphs. JOURNAL OF ALGEBRAIC COMBINATORICS, 62(1) [10.1007/s10801-025-01433-3].
Integral normal Cayley graphs
Spiga P.
2025
Abstract
Given a finite group G, we say that a subset C of G is power-closed if, for every x∈C and y∈⟨x⟩ with ⟨x⟩=⟨y⟩, we have y∈C. In this paper, we are interested in finite Cayley digraphs Cay(G,C) over G with connection set C, where C is a union of conjugacy classes of G. We show that each eigenvalue of Cay(G,C) is integral if and only if C is power-closed. This result will follow from a more general result.
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