Jacobians of Graphs via Edges and Iwasawa Theory

TL;DR

This paper explores the Iwasawa theory of graph Jacobians by constructing an associated Iwasawa module for $Z_p$-towers of graphs, enabling the computation of Jacobian sizes asymptotically.

Contribution

It introduces a new Iwasawa module related to graph Jacobians in $Z_p$-towers and demonstrates its use in asymptotic size computations.

Findings

Constructed an Iwasawa module for graph Jacobians in $Z_p$-towers

Provided examples calculating asymptotic Jacobian sizes

Established connections between graph invariants and Iwasawa theory

Abstract

The Jacobian is an algebraic invariant of a graph which is often seen in analogy to the class group of a number field. In particular, there have been multiple investigations into the Iwasawa theory of graphs with the Jacobian playing the role of the class group. In this paper, we construct an Iwasawa module related to the Jacobian of a $\mathbb{Z}_p$-tower of connected graphs, and give examples where we use this to compute asymptotic sizes of the Jacobians in this tower.