Abstract
We show that if a graph $G$ has average degree $\overline d \geq 4$, 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 $\overline d>4$, 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.
Highlights
Let G = (V, E) denote a graph with vertex set V and edge set E, consisting of unordered pairs of elements of V
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 vertex deck
From the large literature on the subject, we quote the following three sources that are most relevant in the context of our results: (a) vertex-reconstruction of the characteristic polynomial of the vertex adjacency matrix by Tutte [27]; (b) vertex-reconstruction of the the electronic journal of combinatorics 25(2) (2018), #P2.26 number of walks of given length through a given vertex v ∈ V by Godsil and McKay [11]; (c) edge reconstruction for graphs with average degree d 2 log2 |V | by Vladimır Muller [23], improving upon a method of Lovasz [20]
Summary
Two open problems that arise from the proofs and that we want to highlight are the following: (a) can the Ihara zeta function ζG be reconstructed from the (multi-)set {ζG−e : e ∈ E} of Ihara zeta functions of edge-deleted graphs?; (b) for |E| 2, is T semi-simple if and only if G has an end-vertex? The phylogenetic reconstruction problem should be considered in the context of general multigraphs, rather than the more traditional case of trees, and our theorem gives a theoretical underpinning for this more general question of reconstruction
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
