  We show that if a graph $G$ has average degree $\bar d \geq 4$, 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 $\bar d>4$, 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.
  The appendix by Daniel MacDonald established the analogue for multigraphs of
some basic results in reconstruction theory of simple graphs that are used in
the main text.
