Edge reconstruction of the Ihara zeta function

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.Comment: 19 pages, 2 pictures, in version 2 some minor changes and now including an appendix by Daniel McDonal

Rigidity and reconstruction for graphs

The edge reconstruction conjecture of Harary (1964) states that a finite graph G can be reconstructed up to isomorphism from the multiset of its edge-deleted subgraphs G–e (with e running over the edges of G). We put this conjecture in the framework of measure-theoretic rigidity, revealing the importance of the lengths of labeled closed walks for the problem