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 Plus annual

Presents the access of premium content as premium feature

Premium Content

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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $C_t$ be a cycle of length $t$, and let $P_1,\ldots,P_t$ be $t$ polygonchains. A polygon flower $F=(C_t; P_1,\ldots,P_t)$ is a graph obtained byidentifying the $i$th edge of $C_t$ with an edge $e_i$ that belongs to anend-polygon of $P_i$ for $i=1,\ldots,t$. In this paper, we first give anexplicit formula for the sandpile group $S(F)$ of $F$, which shows that thestructure of $S(F)$ only depends on the numbers of spanning trees of $P_i$ and$P_i/ e_i$, $i=1,\ldots,t$. By analyzing the arithmetic properties of thosenumbers, we give a simple formula for the minimum number of generators of$S(F)$, by which a sufficient and necessary condition for $S(F)$ being cyclicis obtained. Finally, we obtain a classification of edges that generate thesandpile group.  Although the main results concern only a class of outerplanar graphs, theproof methods used in the paper may be of much more general interest. We makeuse of the graph structure to find a set of generators and a relation matrix$R$, which has the same form for any $F$ and has much smaller size than that ofthe (reduced) Laplacian matrix, which is the most popular relation matrix usedto study the sandpile group of a graph.

The sandpile group of a polygon flower