The Gonality Sequence of Complete Graphs

Abstract

The gonality sequence $(\gamma_r)_{r\geq1}$ of a finite graph/metric graph/algebraic curve comprises the minimal degrees $\gamma_r$ of linear systems of rank $r$. For the complete graph $K_d$, we show that $\gamma_r = kd - h$ if $r<g=\frac{(d-1)(d-2)}{2}$, where $k$ and $h$ are the uniquely determined integers such that $r = \frac{k(k+3)}{2} - h$ with $1\leq k\leq d-3$ and $0 \leq h \leq k $. This shows that the graph $K_d$ has the gonality sequence of a smooth plane curve of degree $d$. The same result holds for the corresponding metric graphs.

Weierstrass Sets on Finite Graphs

“…Remark 16. The computation of the rank Weierstrass set of complete graphs (Corollary 13) was already implicit in the proof of [6,Theorem 8], a result that gives an upper bound for the gonality sequence of complete graphs. In fact, we note that the rank Weierstrass set of a complete graph coincides with its gonality sequence.…”

Gonality sequences of graphs

“…In this paper, we study the gonality sequence of a graph G, which is the sequence of r th gonalities of G as r ranges from 1 to ∞: gon 1 (G), gon 2 (G), gon 3 (G), gon 4 (G), • • • Recent progress has been made towards determining the gonality sequences of various families of graphs, including the complete graphs K n [10] and the complete bipartite graphs K m,n [8]. Work has also been done to study higher gonalities of Erdös-Renyi random graphs [25].…”

Brill–Noether theory of curves on $\protect \mathbb{P}^1 \times \protect \mathbb{P}^1$: tropical and classical approaches

“…The latter is also the case for smooth curves on P 1 × P 1 . For example, if we fix the bidegree (m, n) = (7,5), then the slope inequality is violated at r = 5 (since d 5 = 17 and d 6 = 21) and at r = 11 (since d 11 = 29 and d 12 = 32).…”