Combinatorial Invariants of Stable Curves in Genus 4: Classification and Computation

This paper focuses on the combinatorial invariants of stable curves by studying their associated pm-graphs on genus 4. These invariants, including (), (), and (), play a key role in the arithmetic intersection theory developed by Zhang and others. We begin by classifying all possible pm-graphs of genus 4 using a combinatorial approach and proving bounds on the number of vertices. Building on this classification, we introduce algorithmic methods to compute the admissible invariants of such graphs. Our approach relies on translating geometric data into graph-theoretic quantities such as edge lengths and vertex valences, and utilizes electrical network analogies like mostly resistance functions and Laplacian matrices to calculate () and (), which are defined in the expressions of the admissible invariants. While our algorithms are applied concretely to genus 4, the methods developed here are adaptable to higher genera and broader families of graphs. This work not only contributes to the computation of local invariants in Zhangs decomposition formula but also lays groundwork for future automated classification and theoretical exploration in arithmetic geometry and its combinatorial counterparts.