Linear series on metrized complexes of algebraic curves

A metrized complex of algebraic curves is a finite metric graph together with a collection of marked complete nonsingular algebraic curves, one for each vertex, the marked points being in bijection with incident edges. We establish a Riemann-Roch theorem for metrized complexes of curves which generalizes both the classical Riemann-Roch theorem and its graph-theoretic and tropical analogues due to Baker-Norine, Gathmann-Kerber, and Mikhalkin-Zharkov. We also establish generalizations of the second author's specialization lemma and its weighted graph analogue due to Caporaso and the first author, showing that the rank of a divisor cannot go down under specialization from curves to metrized complexes. As an application of these considerations, we formulate a generalization of the Eisenbud-Harris theory of limit linear series to semistable curves which are not necessarily of compact type.Comment: Major revision taking into account comments of the referees, e.g., proofs are shortened and clarified, notations improved, changes in organization of the sections, etc. 45 page

A spectral lower bound for the divisorial gonality of metric graphs

Let $\Gamma$ be a compact metric graph, and denote by $\Delta$ the Laplace operator on $\Gamma$ with the first non-trivial eigenvalue $\lambda_1$. We prove the following Yang-Li-Yau type inequality on divisorial gonality $\gamma_{div}$ of $\Gamma$. There is a universal constant $C$ such that $$\gamma_{div}(\Gamma) \geq C \frac{\mu(\Gamma) . \ell_{\min}^{\mathrm{geo}}(\Gamma). \lambda_1(\Gamma)}{d_{\max}},$$ where the volume $\mu(\Gamma)$ is the total length of the edges in $\Gamma$, $\ell_{\min}^{\mathrm{geo}}$ is the minimum length of all the geodesic paths between points of $\Gamma$ of valence different from two, and $d_{\max}$ is the largest valence of points of $\Gamma$. Along the way, we also establish discrete versions of the above inequality concerning finite simple graph models of $\Gamma$ and their spectral gaps.Comment: 22 pages, added new recent references, minor revisio

A transfer principle and applications to eigenvalue estimates for graphs

In this paper, we prove a variant of the Burger-Brooks transfer principle which, combined with recent eigenvalue bounds for surfaces, allows to obtain upper bounds on the eigenvalues of graphs as a function of their genus. More precisely, we show the existence of a universal constants $C$ such that the $k$-th eigenvalue $\lambda_k^{nr}$ of the normalized Laplacian of a graph $G$ of (geometric) genus $g$ on $n$ vertices satisfies $$\lambda_k^{nr}(G) \leq C \frac{d_{\max}(g+k)}{n},$$ where $d_{\max}$ denotes the maximum valence of vertices of the graph. This result is tight up to a change in the value of the constant $C$, and improves recent results of Kelner, Lee, Price and Teng on bounded genus graphs. To show that the transfer theorem might be of independent interest, we relate eigenvalues of the Laplacian on a metric graph to the eigenvalues of its simple graph models, and discuss an application to the mesh partitioning problem, extending pioneering results of Miller-Teng-Thurston-Vavasis and Spielman-Tang to arbitrary meshes.Comment: Major revision, 16 page