Coverings of Graphs and Maps, Orthogonality, and Eigenvectors

  • Authors:

  • Jozef Širáň

  • Affiliations:

  • Department of Mathematics, SvF, Slovak University of Technology, 813 68 Bratislava, Slovakia.siran@lux.svf.stuba.sk

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 2001

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Abstract

Lifts of graph and map automorphisms can be described in terms of voltage assignments that are, in a sense, compatible with the automorphisms. We show that compatibility of ordinary voltage assignments in Abelian groups is related to orthogonality in certain {\cal Z}-modules. For cyclic groups, compatibility turns out to be equivalent with the existence of eigenvectors of certain matrices that are naturally associated with graph automorphisms. This allows for a great simplification in characterizing compatible voltage assignments and has applications in constructions of highly symmetric graphs and maps.