2019/07/19 by Haiyan Chen, Chen, Haiyan, Bojan Mohar +1 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Theoretical and Computational Physics #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.1907.08450
openalex publication_date 2019/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Ct be a cycle of length t, and let P1,…,Pt be t polygon chains. A polygon flower F=(Ct; P1,…,Pt) is a graph obtained by identifying the ith edge of Ct with an edge ei that belongs to an end-polygon of Pi for i=1,…,t. In this paper, we first give an explicit formula for the sandpile group S(F) of F, which shows that the structure of S(F) only depends on the numbers of spanning trees of Pi and Pi/ ei, i=1,…,t. By analyzing the arithmetic properties of those numbers, 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 cyclic is obtained. Finally, we obtain a classification of edges that generate the sandpile group. Although the main results concern only a class of outerplanar graphs, the proof methods used in the paper may be of much more general interest. We make use 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 of the (reduced) Laplacian matrix, which is the most popular relation matrix used to study the sandpile group of a graph.