Some Results On The Topological Indices Of Phenylene Chains

The research of topological properties of polycyclic hydrocarbons has a long history, thereby it has promoted the research on some chemical graphs such as poly-omino chains, hexagonal chains and so on. So far, many results have been achieved on them. The mathematical research about them mainly focuses on tiling problem, enumerations, matchings counting, independent sets counting and respective ordering problem, etc. In this article, we will consider another special chemical graph: phenylene chains.The dissertation includes two chapters. The fist chapter is introduction. In the second chapter, we discuss the extremal phenylene chains on k-matchings and k-independent sets. Denote by P_n the set of the phenylene chains with n hexagons. For any PH_n∈P_n, let m_k(PH_n) and i_k(PH_n) be the number of k-matchings and k-independent sets of PH_n, respectively. In this chapter, we show that for any PH_n∈P_n and any k≥0, m_k(L_n)≤m_k(PH_n)≤m_k(H_n) and i_k(L_n)≥i_k(PH_n)≥i_k(H_n), with the left of equalities holding for all k only if PH_n = L_n, the right equalities holding for all k only if PH_n = H_n, where L_n and H_n are the linear chain and the helical chain, respectively. Futhermore, we also discuss the extremal phenylene chains concerning Hosoya index, Merrifield-Simmons index and phenylene chains with respect to totalπ-electron energy.