Some Results On Random Polyomino Chains

This thesis falls into 4 parts.In chapter 1,we introduce some basic concepts and symbols used in this thesis,and make a brief introduction of the background and development of the related fields.In chapter 2 and chapter 3,we obtain the explicit expressions of the expectation of the number of perfect matchings,Hosoya index,Wiener index and hyper-Wiener index on random polyomino chains.Based on those formulae,the corresponding index of linear chain and zig-zag chain can be obtained respectively as the probability parameter p takes different values.In the last chapter,we proved that for any square-hexagonal chain R_n,there exists a corresponding caterpillar tree T_nsuch that the number of perfect matchings of R_nis equal to the Hosoya index of T_n,which extends the result of hexagonal chain obtained by Gutman(Topological properties of benzeoid systems,Theoret.Chim.Acta,45(1977),307-315)and the result of polyomino chain obtained by Shuli Li and Weigen Yan(Kekul′e structures of polyomino chains and the Hosoya index of caterpillar trees,Discrete Mathematics,(2012),2397-2400).