The Sandpile Group of Polygon Flower with Two Centers

Abstract

Let $C_{k+1}$ be a cycle of length k + 1 and $C_{t+1}$ be a cycle of length t + 1. A polygon flower with two centers, denoted by $F=F(C_{k+1};P_1,\dots ,P_k;C_{t+1};P_{k+1},\dots ,P_{k+t})$ is obtained by identifying the $i\text{th}$ edge of $C_{k+1}$ with an edge e_i that belongs to an end-polygon of P_i for $i=1,\dots ,k,$ and identifying the $j\text{th}$ edge of $C_{t+1}$ with an edge e_j that belongs to an end-polygon of P_j for $j=k+1,\dots ,k+t,$ where $C_{k+1}$ and $C_{t+1}$ have a common edge h. In this paper, we determine the order of sandpile group S(F) of F, which can be viewed as generalized of results in paper (Haiyan Chen, Bojan Mohar. The sandpile group of a polygon flower. Discrete Applied Mathematics, 2019). Moreover, the formula and structure for sandpile group of polygon flower can be obtained. Finally, as application of our result, we also present the sandpile group of cata-condensed system with two branched hexagons.

Keywords:

• Sandpile group
• Abelian group
• polygon flower

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by NSFC (Grant nos.11761070, 61662079). 2021 Xinjiang Uygur Autonomous Region National Natural Science Foundation Joint Research Fund (2021D01C078), 2020 Special Foundation for First-class Specialty of Applied Mathematics Xinjiang Normal University.