English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Let $G$ be a connected graph. The Jacobian group (also known as the Picardgroup or sandpile group) of $G$ is a finite abelian group whose cardinalityequals the number of spanning trees of $G$. The Jacobian group admits acanonical simply transitive action on the set $\mathcal{G}(G)$ of cycle-cocyclereversal classes of orientations of $G$. Hence one can construct combinatorialbijections between spanning trees of $G$ and $\mathcal{G}(G)$ to buildconnections between spanning trees and the Jacobian group. The BBY bijectionsand the Bernardi bijections are two important examples. In this paper, weconstruct a new family of such bijections that include both. Our bijectionsdepend on a pair of atlases (different from the ones in manifold theory) thatabstract and generalize certain common features of the two known bijections.The definition of these atlases is derived from triangulations and dissectionsof the Lawrence polytopes associated to $G$. The acyclic cycle signatures andcocycle signatures used to define the BBY bijections correspond to regulartriangulations. Our bijections can extend to subgraph-orientationcorrespondences. Most of our results hold for regular matroids. We present ourwork in the language of fourientations, which are a generalization oforientations.

A framework unifying some bijections for graphs and its connection to Lawrence polytopes