Abstract. We present a method to control gonality of nonarchimedean curves based on graph theory.Let k denote a complete nonarchimedean valued field. We first prove a lower bound for the gonality of a curve over the algebraic closure of k in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve.Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some "volume" of the original graph; this can be seen as a substitute for graphs of the Li-Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs.Finally, we apply the results to give a lower bound for the gonality of arbitrary Drinfeld modular curves over finite fields and for general congruence subgroups Γ of Γ(1) that is linear in the index [Γ(1) : Γ], with a constant that only depends on the residue field degree and the degree of the chosen "infinite" place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian.

Abstract. Let Γ be a compact metric graph, and denote by ∆ the Laplace operator on Γ with the first non-trivial eigenvalue λ1. We prove the following Yang-Li-Yau type inequality on divisorial gonality γ div of Γ. There is a universal (explicit) constant C such thatwhere the volume µ(Γ) is the total length of the edges in Γ, ℓ geo min is the non-zero minimum length of all the geodesic paths between points of Γ of valence different from two, and dmax is the largest valence of points of Γ. Along the way, we also establish discrete versions of the above inequality concerning finite simple graph models of Γ and their spectral gaps.

Abstract. We construct explicitly an infinite family of Ramanujan graphs which are bipartite and biregular. Our construction starts with the Bruhat-Tits building of an inner form of SU 3 (Qp). To make the graphs finite, we take successive quotients by infinitely many discrete co-compact subgroups of decreasing size.

One can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincaré disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic.In this paper, we find the corresponding statement for Mumford curves, a nonarchimedean analog of Riemann surfaces. In this case, the mere absolute continuity of the boundary map (for Schottky uniformization and the corresponding Patterson-Sullivan measure) only implies isomorphism of the special fibers of the Mumford curves, and the absolute continuity needs to be enhanced by a finite list of conditions on the harmonic measures on the boundary (certain nonarchimedean distributions constructed by Schneider and Teitelbaum) to guarantee an isomorphism of the Mumford curves.The proof combines a generalization of a rigidity theorem for trees due to Coornaert, the existence of a boundary map by a method of Floyd, with a classical theorem of Babbage-Enriques-Petri on equations for the canonical embedding of a curve.

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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".

We show that if a graph G has average degree d ≥ 4, then the Ihara zeta function of G is edgereconstructible. 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 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.Date: August 22, 2018 (version 2.0). 2010 Mathematics Subject Classification. 05C50, 05C38, 11M36, 37F35, 53C24.A degree-one vertex is called an end-vertex. All results in this paper hold for connected finite undirected multigraphs without end-vertices, and from now on we will use the word "graph" for such multigraphs.If e = {v 1 , v 2 } ∈ E, we denote by e = (v 1 , v 2 ) the edge e with a chosen orientation, and by e = (v 2 , v 1 ) the same edge with the inverse orientation to that of e. Let o( e) = v 1 denote the origin of e and t( e) = v 2 its end point. If there are multiple edges between v 1 , v 2 then we will label them e i = (v 1 , v 2 ) i . A nonbacktracking edge walk of length n is a sequence e 1 e 2 ....e n of edges such that t(e i ) = o(e i+1 ), but e i+1 = e i . We call it tailless if e n = e 1 . Just like walks in the graph can be studied using the adjacency matrix, nonbacktracking walks are captured by the edge adjacency matrix T = T G studied by Sunada [24], Hashimoto [14] and Bass [2]. Letting E denote the set of oriented edges of G for any possible choice of orientation, so | E | = 2|E|, T is defined to be the 2|E| × 2|E| matrix, in which the rows and columns are indexed by E, and T e 1 , e 2 = 1 if t( e 1 ) = o( e 2 ) but e 2 = e 1 ; 0 otherwise.

Funding: The funding that lead to this publication stems from two KB projects; KB-38-001-005 Autonomous Collaborative Robots, part of Data Driven High Tech programme (JK), and KB36-2C4 Marine lower trophic food systems (MB).