Edge reconstruction of the Ihara zeta function

Abstract

We show that if a graph G has average degree d ≥ 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 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.