Let $$C_{k_1}, \ldots , C_{k_n}$$ be cycles with $$k_i\ge 2$$ vertices ( $$1\le i\le n$$ ). By attaching these n together in a linear order, we obtain graph called polygon chain. cyclic graph, which is ring if it can embedded on the plane; and twisted Möbius band. It known that sandpile group of chain always cyclic. Furthermore, there exist edge generators. In this paper, not only show any (twi...

when addition and multiplication are defined in the obvious way, form a ring, the group-ring of G over K, which will be denoted by R (G, K). Henceforward, we suppose that K has the modulus 1, and we denote the identity in G by e0. Then R(G,K) has the modulus l.e0. Since no confusion can arise thereby, the element 1. e in R(G, K) will be written as e, and whenever it is convenient, the elements ...