The Extremal Numbers Of Kekulé Structures Of Polygonal Chains

A perfect-matching(i.e,1-factor)is a matching which matches all vertices of the graph.A Kekul?e structures of an aromatic compound consists of a perfect matching of carbon skeleton,showing the locations of double bonds in the chemical structure.It plays a central role in theory of chemical structure.The enumeration of Kekul?e structures is traditional and extensively elaborated field of mathematical chemistry.In 1977,Ivan Gutman discovered that the Kekul?e structure count of a hexagonal chain is equal to Hosoya index of corresponding caterpillar.We determine the first few maximal and minimal values of Hosoya index of caterpillars and characterize the extremal graphs,and then we give the extremal numbers of Kekul?e structures of some polygonal chains and the corresponding graphs by using the relations between Hosoya index of caterpillars and the number of Kekul?e structures of polygonal chains.This paper consists of four parts.Part 1 is preface and preliminaries.In part 2,we determine the first ten minimal and the first five maximal Hosoya index and characterize the extremal graphs.In part 3,we determine the extremal numbers of Kekul?e structures in hexagonal chains,polyomino chains,square-hexagonal chains,pentagonal chains and characterize the corresponding extremal graphs.In part 4,we give a formula for computing the number of Kekul?e structures of square-hexagonal alternating spider chain.