Zeta functions of edge-free quotients of graphs

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We define the Ihara zeta function ζ(u,X) and Artin–Ihara L-function of the quotient graph of groups X=Y//G, where G is a group acting on a finite graph Y with trivial edge stabilizers. We determine the relationship between the primes of Y and X and show that the projection map Y→X can be naturally viewed as an unramified Galois covering of graphs of groups. We show that the L-function of X evaluated at the regular representation is equal to ζ(u,Y), and that ζ(u,X) divides ζ(u,Y). We derive two-term and three-term determinant formulas for the zeta and L-functions, and compute several examples of L-functions of edge-free quotients of the tetrahedron graph K4.

Original languageEnglish
Pages (from-to)40-71
Number of pages32
JournalLinear Algebra and Its Applications
StatePublished - Nov 15 2021


  • Graph of groups
  • Ihara zeta function
  • L-function


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