Non-Isomorphic Cayley Graphs of Metacyclic Groups of Order 8 p with the Same Spectrum

Lang Tang, Weijun Liu and Rongrong Lu ()
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Lang Tang: School of Mathematics and Statistics, Hunan First Normal University, Changsha 410205, China
Weijun Liu: School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410083, China
Rongrong Lu: School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha 410083, China

Mathematics, 2025, vol. 13, issue 12, 1-14

Abstract: The spectrum of a graph Γ , denoted by S p e c ( Γ ) , is the multiset of eigenvalues of its adjacency matrix. A Cayley graph C a y ( G , S ) of a finite group G is called Cay-DS (Cayley graph determined by its spectrum) if, for any other Cayley graph C a y ( G , T ) , S p e c ( C a y ( G , S ) ) = S p e c ( C a y ( G , T ) ) implies C a y ( G , S ) ≅ C a y ( G , T ) . A group G is said to be Cay-DS if all Cayley graphs of G are Cay-DS. An interesting open problem in the area of algebraic graph theory involves characterizing finite Cay-DS groups or constructing non-isomorphic Cayley graphs of a non-Cay-DS group that share the same spectrum. The present paper contributes to parts of this problem of metacyclic groups M 8 p of order 8 p (with center of order 4), where p is an odd prime, in terms of irreducible characters, which are first presented. Then some new families of pairwise non-isomorphic Cayley graph pairs of M 8 p ( p ≥ 5 ) with the same spectrum are found. As a conclusion, this paper concludes that M 8 p is Cay-DS if and only if p = 3 .

Keywords: metacyclic groups; Cayley graphs; isospectral graphs; irreducible characters (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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