Extremal Merrield-Simmons Index And Hosoya Index Of Six-membered Ring Spiro Spiders And Polyphenyl Spiders

Let G= (V, E) be a simple, undirected graph. If each vertex of G represents an atom of molecule and each edge represents the chemical bond between the atoms respectively, then G is called an molecular graph. It is well known that the appearance and the de-velopment of graph theory are closely connected 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 areas in the modern chemical graph theory. For some topolog-ical properties of the chemical molecular graph, many results have been achieved. The mathematical research about them mainly focuses on tiling problem, enumeration, match-ings counting, independent sets counting and related ordering problem, etc. In chemical graph theory, independent numbers(called Merrifield-Simmons index in the chemistry), matching numbers(called Hosoya index in the chemistry)are two widely use of topological indices. For the two indices, there had been some results on six-membered ring spiro chains and Polyphenyl chains. In this paper, we extend the main results to a kind of more general graphs containing branched six-membered ring multispiro molecules and multipolyphenyl molecules(in this paper we called them six-membered ring spiro spiders and Polyphenyl spiders, respectively).This thesis consists of three chapters. The first chapter is divided into two sections. In the first section, we introduce basic concept, terminology and symbol. In section 2, we firstly introduce research background of Merrifield-Simmons index and Hosoya index, then roundup the work had been done about this paper. The second chapter we determine the six-membered spiro ring spiders having extremal values of Merrifield-Simmons index and Hosoya index, respectively. The third chapter we define Polyphenyl spiders by Polyphenyl chain based on the second chapter, then determine the Polyphenyl spiders having extremal values of Merrifield-Simmons index and Hosoya index, respectively.Here are main results in this paper: 1.We use Fn1,n2,n3 to denote the set of all six-membered ring spiro spiders (?)(n1,n2,n3)with three legs of lengths n1,n2 and n3.We have proofed that for (?)(n1,n2,n3)(?)Fn1,n2,n3, if m=n1+n2+n3=n'1+n'2+n'3≥5,satisfying n'1≤n'2≤n'3 and n'3-n'1≤1.Thenσ((?)R(2,2,m-4))≤σ(6(n1,n2,n3))≤σ((?)s(n'1,n'2,n'3)).2.We use Fn1.n2,n3 to denote the set of all six-membered ring spiro spiders 6(n1,n2,n3)with three legs of lengths n1,n2 and n3.We have proofed that for (?)(n1,n2,n3)(?)Fn1,n2,n3, if m=n1+n2+n3=nn'1+n'2n'3≥5,satisfying n'1≤n'2≤n'3 and n'3-n'1≤1.Then z((?)s(n'1:,n'2,n'3))≤z((?)(n1,n2,n3))≤z((?)R(2,2,m-4)).3. We use (?)n1,n2,n3 to denote the set of all Polyphenyl spiders 6(n1,n2,n3)with three legs of lengths n1,n2 and n3.We have proofed that for (?)(n1,n2,n3)(?)n1,n2,n3,set m=n1+n2+n3≥3.Thenσ((?)R1(1,1,m-2))≤σ((?)(n1,n2,n3))≤σ((?)s3(1,1,m-2)).4. We use (?)n1,n2,n3 to denote the set of all Polyphenyl spiders 6(n1,n2,n3)with three legs of lengths n1,n2 and n3.We have proofed that for (?) (n1,n2,n3)(?)n1,n2,n3,set m=n1+n2+n3≥3.Then z((?)s3≥(1,1,m-2))≤z((?)(n1,n2,n3))≤z((?)R1(1,1,m一2)).