The Jacobian of a graph and graph automorphisms

TL;DR

This paper explores the relationship between graph automorphisms and the Jacobian, revealing conditions under which the Jacobian cannot be cyclic, especially for certain highly connected graphs with nonabelian automorphism groups.

Contribution

It demonstrates that for 3-edge-connected graphs with nonabelian automorphism groups, the Jacobian cannot be cyclic, advancing understanding of the Jacobian's combinatorial properties.

Findings

Nonabelian automorphism groups imply non-cyclic Jacobians for certain graphs

Cayley graphs of degree ≥3 from nonabelian groups have non-cyclic Jacobians

Provides new insights into the combinatorial interpretation of the Jacobian's rank

Abstract

In the present paper we investigate the faithfulness of certain linear representations of groups of automorphisms of a graph $X$ in the group of symmetries of the Jacobian of $X$. As a consequence we show that if a $3$-edge-connected graph $X$ admits a nonabelian semiregular group of automorphims, then the Jacobian of $X$ cannot be cyclic. In particular, Cayley graphs of degree at least three arising from nonabelian groups have non-cyclic Jacobians. While the size of the Jacobian of $X$ is well-understood - it is equal to the number of spanning trees of $X$ - the combinatorial interpretation of the rank of Jacobian of a graph is unknown. Our paper presents a contribution in this direction.