Edge reconstruction of the Ihara zeta function

Gunther Cornelissen, Janne Kool

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We show that if a graph G has average degree (d)over-bar >= 4, then the Ihara zeta function of G is edge-reconstructible. We prove some general spectral properties of the edge adjacency operator T: it is symmetric for an indefinite form and has a "large" semi-simple part (but it can fail to be semi-simple in general). We prove that this implies that if -dover-bar > 4, one can reconstruct the number of non-backtracking (closed or not) walks through a given edge, the Perron-Frobenius eigenvector of T (modulo a natural symmetry), as well as the closed walks that pass through a given edge in both directions at least once.
Original languageEnglish
Article numberP2.26
JournalThe Electronic Journal of Combinatorics
Issue number2
Number of pages22
Publication statusPublished - 2018


  • Edge reconstructing conjecture
  • Ihara zeta function
  • Non-backtracking walks


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