[Objective]In this paper,we study the problem of extremal phenylene chains with n hexagons on k-independent sets.[Methods]We show the partial order relationship among the linear chain,phenylene chain and helical chain by the recursion relation of Y-polynomial of phenylene chain and the inducion method.Then we use the partial order relationship of phenylene chains to determine extremal phenylene chains on k-independent sets.[Results]We show that the maximal chain on k-independent sets is a linear chain,and the minimal chain is a helical chain.In addition,we also determine the recursion relation of Y-polynomial of phenylene chain.[Conclusion]To determine extremal phenylene chains with n hexagons on k-independent sets,we try to find the recursion relation of Y-polynomial of phenylene chains.Hopefully,our study may contribute to subsequent studies such as Merrifield-Simmons index,and determine the partial order relationship of Y-polynomial of phenylene chains.
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