Edge Reconstruction of the Ihara Zeta Function

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 Bass–Hashimoto edge adjancency operator T : it is symmetric on a Kreı̆n space 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. Introduction Let G = (V,E) denote a graph with vertex set V and edge set E ⊆ (V × V )/S2, consisting of unordered pairs of elements of V . The edge deck De(G) of G is the multi-set of isomorphism classes of all edge-deleted subgraphs of G. Harary [13] conjectured in 1964 that graphs on at least four edges are edge-reconstructible, i.e., determined up to isomorphism by their edge deck. This so-called edge reconstruction conjecture is the analogue for edges of the famous (vertex) reconstruction conjecture of Kelly and Ulam that every graph on at least three vertices is determined by its (similarly defined) vertex deck (compare [4]). Many invariants of graphs where shown to be reconstructible from the vertex and/or edge deck, and from the large literature on the subject, we quote the following three sources that are most relevant in the context of our results: (a) vertexreconstruction of the characteristic polynomial (of the vertex adjacency matrix) by Tutte [25]; (b) vertex-reconstruction of the number of (possibly backtracking) walks of given length through a given vertex v ∈ V (which one can specify without knowing the graph G by pointing to the element G − v of the vertex deck) by Godsil and McKay [11]; (c) edge reconstruction for graphs with average degree d ≥ 2 log2 |V | by Vladimír Müller [22], improving upon a method of Lovász [19]. Following the discussion by McDonald in [20], the edge reconstruction conjecture should also hold for multi-graphs. Since disconnected (multi-)graphs are reconstructible ([4], 6.14(b), [20], Date: July 14, 2015 (version 1.0). 2010 Mathematics Subject Classification. 05C50, 05C38, 11M36, 37F35, 53C24.