Rigidity and reconstruction for graphs

Abstract

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".

Edge Reconstruction of the Ihara Zeta Function

“…We finish this introduction by listing some applications. As we explain in [7] (cf. also [9]), the invariants that we have reconstructed play a central role in the measure-theoretical study of the action of the fundamental group on the boundary of the universal covering tree of the graph.…”

“…Notice that p and q are determined up to scaling, but different choices do not change the right hand side of the equation. Now Formula (7) implies the following equivalence between left and right eigenvectors v for T :…”

Komunikacja kryzysowa jako czynnik rozwoju kryzysu

“…Istnieje wiele przykładów na to, że źródłem sytuacji kryzysowej są już nie tyle realne działania firm czy ich pracowników, ile źle przygotowane akty komunikacji, wywołujące oburzenie i sprzeciw odbiorców. Kryzysy wywołane przez niewłaściwą komunikację mogą realnie wpłynąć na funkcjonowanie firmy czy instytucji (partii politycznej, rządu), poprzez bojkot ze strony konsumentów czy spadek poparcia, a w niektórych przypadkach zagrozić nawet istnieniu organizacji 22 .…”