Related papers: Hyperelliptic graphs and metrized complexes

In an influential 2008 paper, Baker proposed a number of conjectures relating the divisor theory of algebraic curves with an analogous combinatorial theory on finite graphs. In this note, we examine Baker's Brill--Noether existence…

Classes of graphs with bounded expansion are a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r,…

This note uses a variation of graded Morita theory for finite dimensional superalgebras to determine explicitly the graded basic superalgebras for all real and complex Clifford superalgebras. As an application, the Grothendieck groups of…

This work investigates the existence of complex structures on 2-step nilpotent Lie algebras arising from finite graphs. We introduce the notion of adapted complex structure, namely a complex structure that maps vertices and edges of the…

We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized…

We consider hereditary classes of graphs equipped with a total order. We provide multiple equivalent characterisations of those classes which have bounded twin-width. In particular, we prove a grid theorem for classes of ordered graphs…

A hypergraph is said to be $1$-Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of $1$-Sperner hypergraphs and their structure to graphs. In particular, we…

Recently, Baker and Norine have proven a Riemann-Roch theorem for finite graphs. We extend their results to metric graphs and thus establish a Riemann-Roch theorem for divisors on (abstract) tropical curves.

In this paper we construct a class of bounded degree bipartite graphs with a small separator and large bandwidth. Furthermore, we also prove that graphs from this class are spanning subgraphs of graphs with minimum degree just slightly…

We study the locus of tropical hyperelliptic curves inside the moduli space of tropical curves of genus g. We define a harmonic morphism of metric graphs and prove that a metric graph is hyperelliptic if and only if it admits a harmonic…

Proves that if a curve has totally split reduction and the corresponding skeleton (as a metric graph) is a chain of loops with generic edge lengths, then every divisor on the graph with imposed ramification at the rightmost vertex $p$ of…

We introduce the notion of k-hyperclique complexes, i.e., the largest simplicial complexes on the set [n] with a fixed k-skeleton. These simplicial complexes are a higher-dimensional analogue of clique (or flag) complexes (case k=2) and…

Let $(X,H)$ be a polarized K3 surface with $\mathrm{Pic}(X) = \mathbb Z H$, and let $C\in |H|$ be a smooth curve of genus $g$. We give an upper bound on the dimension of global sections of a semistable vector bundle on $C$. This allows us…

This paper supplements [17], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories,…

A metric graph is a geometric realization of a finite graph by identifying each edge with a real interval. A divisor on a metric graph $\Gamma$ is an element of the free abelian group on $\Gamma$. The rank of a divisor on a metric graph is…

We give bounds on the L(2,1)-labeling number of a simple graph in terms of its order and its maximum degree. We also describe an infinite class of graphs of which the elements have the highest L(2,1)-labeling numbers in terms of their…

Given a finite set of points in $\mathbb{R}^d$, Tverberg's theorem guarantees the existence of partitions of this set into parts whose convex hulls intersect. We introduce a graph structured on the family of Tverberg partitions of a given…

Intuitively speaking, a bipartite graph is mirror if it can be drawn in the Cartesian plane in such a way that, the vertices of one stable are points in x=0, the vertices of the other stable set are points in x=1, the edges are straight…

A graded tensor category over a group $G$ will be called a strongly $G$-graded tensor category if every homogeneous component has at least one multiplicativily invertible object. Our main result is a description of the module categories…

We introduce a notion of matrix valued Gram decompositions for correlation matrices whose study is motivated by quantum information theory. We show that for extremal correlations, the matrices in such a factorization generate a Clifford…