Clifford's Theorem for graphs

TL;DR

This paper extends Clifford's Theorem to metric graphs, showing that certain linear systems imply the existence of a degree 2 linear system, paralleling classical results for algebraic curves.

Contribution

It establishes a Clifford-type theorem for metric graphs, connecting the existence of specific linear systems to the presence of a degree 2 system.

Findings

If a metric graph has a linear system g^r_{2r} with 2 ≤ r ≤ g-2, then it also has a g^1_2.

The result parallels classical Clifford's Theorem for smooth projective curves.

Provides a combinatorial analogue of a fundamental algebraic geometry theorem.

Abstract

Let $\Gamma$ be a metric graph having a linear system $g^r_{2r}$ for some $2 \leq r \leq g-2$ then $\Gamma$ has a linear system $g^1_2$. This is similar to the well-known Clifford's Theorem from the theory of linear systems on smooth projective curves.