The Sandpile Group of Polygon Flower with Two Centers

Let Ck+1 be a cycle of length k + 1 and Ct+1 be a cycle of length t + 1. A polygon flower with two centers, denoted by F = F (Ck+1; P-1, ..., P-k; Ct+1; Pk+1, ..., Pk+t) is obtained by identifying the ith edge of Ck+1 with an edge e(i) that belongs to an end-polygon of P-i for i = 1, ..., k, and identifying the jth edge of Ct+1 with an edge e(j) that belongs to an end-polygon of P-j for j= k + 1, ..., k + t, where Ck+1 and Ct+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.