Edge Reconstruction of the Ihara Zeta Function

Cornelissen, Gunther; Kool, Janne

doi:10.37236/5909

Cited by 2 publications

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“…The average degree d of G then equals d = v∈V deg v/|V | = 2|E|/|V |. In [6], we have proven the following: Looking at Theorems 3.2 and 4.1 simultaneously indicates that, from this point of view, reconstruction is intimately related to knowledge about lengths of closed walks.…”

Section: Reconstruction Of Measure-theoretic Invariantsmentioning

confidence: 83%

Rigidity and reconstruction for graphs

Cornelissen

Kool

2019

J. Fractal Geom.

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We present measure theoretic rigidity for graphs of first Betti number b > 1 in terms of measures on the boundary of a 2b-regular tree, that we make explicit in terms of the edge-adjacency and closed-walk structure of the graph. We prove that edge-reconstruction of the entire graph is equivalent to that of the "closed walk lengths". Some rigidity phenomenaA compact Riemann surface X of genus g ≥ 2 is uniquely determined by a dynamical system, namely, the action of the fundamental group Π g in genus g on the Poincaré disk ∆ by Möbius transformations. Things change when we replace ∆ by its real one-dimensional boundary ∂∆ = S 1 ; the action of Π g extends to S 1 , but this action will only depend on the topological isomorphism type of X, viz., the genus g. Rigidity re-enters the picture via the Lebesgue-measure on S 1 , in the sense that two Riemann surfaces X and Y are isomorphic if and only if there exists a Π g -equivariant absolutely continuous homeomorphism S 1 → S 1 (cf. e.g. [8]).A similar result holds for more general hyperbolic spaces. We describe a version for graphs (cf. Coornaert [3]): let G = (V, E) denote a graph with vertex set V and edge set E, consisting of unordered pairs of elements of V . Let b denote the first Betti number of G, and assume b ≥ 2.Knowing G is the same as knowing the action of a free group F b or rank b on the universal covering tree T of G. Again, the dynamical system of F b acting on the boundary ∂T of T (i.e., the space of ends of T) is topologically conjugate to a system that only depends on b (to wit, the action of F b on the boundary of its Cayley graph), but if one considers the set of Patterson-Sullivan measures on ∂T, rigidity holds; we provide an exact result in Theorem 2.3 below.The graph rigidity theorem shows the importance of the structure of the "space of closed walks" (C, F b , µ) of a graph in understanding the structure of a graph. We apply this insight to reconstruction problems for graphs. We find that for average degree > 4, we can reconstruct various ingredients of the explicit formula for the rigidifying measure. We conclude that reconstruction of a graph is intimately related with the structure of lengths of closed walks in a graph. The final section confirms this; we prove in an elementary way that the edge reconstruction conjecture is equivalent to the reconstruction of "closed walks and their length".

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