Some Indices Of Biphenyl And Polycyclic Hydrocarbons

Let G= (V, E) be a simple, undirected graph. If each vertex of G repre-sents an atom of a molecule and each edge represents a chemical bond between the atoms respectively, then G is called an molecular graph. It is well known that the appearance and the development of graph theory are closely connect-ed with the research of chemical molecular graph. The study of molecular topological indices and the invariants of molecular graph is one of the most active area in the modern chemical graph theory. A molecular topological in-dex is from the set of molecular graphs to the set of real number that is each real number represents a molecular graph. For some topological properties of chemical molecular graph, many results have been achieved. The research of graph theory about them mainly focuses on theory of graph spectra, theory of extremal graph, matchings counting, counting problem, etc.In this paper, we focus on the extremal graphs and ordering problems respect to some topological indices, we will mainly discuss the matching ener-gy, Kirchoff-index, Hosoya-index and Merrifleld-Simmons-index of polyphenyl chains and spiro chains.Here is the main work in this paper:1. matching energy In chapter two and chapter three, firstly, we study the matching energy of polyphenyl chains and spiro chains by some transformation-s of graphs, auxiliary graphs and inductive recursion. We show that if PPCn is a polyphenyl chain of length n, then we have ME(Mn)< ME(PPCn)< ME(On). We also study the order respect to matching energy of polyphenyl chains, and get the extremal graphs of the second minimal, the second maxi-mal matching energy in the polyphenyl chains of length n. Secondly, we study the matching energy of spiro chains by some transformations of graphs, auxil-iary graphs. We get that if SPCn is a six-membered ring spiro chain of length n, then we have ME(Mn)< ME(SPCn)< ME(On). We also get the ex-tremal graphs of the minimum, the second minimal, the maximum, the second maximal matching energy of the six-membered ring spiro chains of length n.2. Kirchhoff-index In chapter four, by using the properties of Kirchoff- index,we get the expected value of Kirchoff-index of raildom polyphenyl chains PPC(n,p1,p2)is E(Kf(PPC(n,p1,p2)))=(15-p1-4p2)n3+(3p1+12p2+8)n2-(2p1+8p2+11/2)n and the expected value of Kirchoff-index of random six-membered ring spiro ch ains SPC(n,p1,p2)is E(Kf(SPC(n,p1,p2)))=(25/4-25/36p1-25/9p2)n3+(25/12p1+25/3p2+125/12)n2-(25/18p1+50/9p2-5/6)n.3.Hosoya-index In chapter five,by using the properties of Hosoya-index and some auxiliary graphs:we get the expected value of Hosoya-index of ran-dom polyphenyl chains PPC(n,p1,p2)is and the expected value of Hosoya-index of random six-membered ring spiro chains SPC(n,p1,p2)is4. Merrifield-Simmons-index In chapter six,by using the properties of Merrifield-Simmons-index and some auxiliary graphs,we get the expected val-ue of Merrifield-Simmons-index of random polyphenyl chains and the expected value of Merrifield-Simmons-index of random six-membered ring spiro chains SPC(n,p1,p2) is...