The Jacobian of Cyclic Voltage Covers of $K_n$

TL;DR

This paper investigates the structure and order of Jacobians of cyclic voltage covers of complete graphs, providing explicit formulas and structural insights for these derived graphs using algebraic and zeta-function techniques.

Contribution

It determines the Jacobian's order and structure for single voltage covers of complete graphs, introducing explicit formulas and new connections via zeta-functions.

Findings

Explicit formulas for Jacobian order of derived graphs

Structural characterization of Jacobians for cyclic voltage covers

Product formulas relating Jacobians of base and covering graphs

Abstract

This paper proves results about the Jacobians of a certain family of covering graphs, $Y$, of a base graph $X$, that is constructed from an assignment of elements from a group $G$ to the edges of $X$ ($G$ is called the voltage group and $Y$ is called the derived graph). Of particular interest is when the voltage assignment is given by mapping a generator of the cyclic group of order $d$ to a single edge of $X$ (all other edges are assigned the identity), called a single voltage assignment. Both the order and abelian group structure of the Jacobian of single voltage assignment derived graphs are determined when the base graph $X$ is the complete graph on $n$ vertices, for every $n$ and $d$. Using zeta-functions, general product formulas that relate the order of the Jacobian of $Y$ to that of $X$ are developed; these formulas become very simple and explicit in the special case of single voltage covers of $X$.