Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groups

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Dalfó, CristinaDalfó, Cristina - ORCID ID
Fiol, Miguel Angel
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cc-by-nc-nd (c) Elsevier, 2020
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In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on n letters. We prove that every partition of the number n gives rise to a regular partition of the Cayley graph. By using representation theory, we also obtain the complete spectra and the eigenspaces of the corresponding quotient (di)graphs. More precisely, we provide a method to find all the eigenvalues and eigenvectors of such (di)graphs, based on their irreducible representations. As examples, we apply this method to the pancake graphs P(n) and to a recent known family of mixed graphs Γ(d, n, r) (having edges with and without direction). As a byproduct, the existence of perfect codes s in P(n) allows us to give a lower bound for the multiplicity of its eigenvalue −1.
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Linear Algebra and its Applications, 2020, vol. 597, p. 94-112