Some Extremal Problems On The Topological Indices Of Polygonal Chains

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 development 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 area in the modern chemical graph theory. For some topological properties of the chemical molecular graph, many results have been achieved. The mathematical research about them mainly focuses on tiling problem, enumerations, matchings counting, independent sets counting and related ordering problem, etc.In graph theory, matching numbers (called Hosoya index in the chemistry), independent numbers (called Merrifield- Simmons index in the chemistry) and Wiener index are three important parameter. They have significant applied background and widely use in chemical graph theory. It is naturally to consider the extremal problem with respective to these topological indices. For first and second indices, [47] and [2] had given some results on polyomino chains and hexagonal chains. For Wiener index, [67] had given some results on hexagonal chains. In this thesis, we extend the main results to a more general case.Here are main results in this paper:In chapter 2, we discuss the extremal pentagonal chains on k-matchings and k-independent sets. Denote by A_n the set of the pentagonal chains with n pentagons. For any pentagonal chain A_n∈A_n,let m_k(A_n) and i_k(A_n) be the number of k-matchings and k-independent sets of A_n,respectively. In the second chapter, we show that for any A_n∈A_n and any k≥0,m_k(Z_n~2)≤m_k(A_n)≤m_k(Z_n~1) and i_k(Z_n~2)≥i_k(A_n)≥(Z_n~1),with the left of equalities holding for all k only if A_n=Z_n~2;the right of equalities holding for all k only if A_n= Z_n~1,where Z_n~1 and Z_n~2 are the chain of type one and the chain of type two, respectively (see figure 4 (a) and 4 (b)). In chapter 3, we further discuss the extremal h-polygonal chains (h>5) on k-matchings and k-independent sets. Denote by A_n the set of the polygonal chains with n polygons. For any A_n∈A_n,let m_k(A_n) and i_k(A_n) be the number of k-matchings and k-independent sets of A_n,respectively. In the third chapter, we show that for any polygonal chain A_n∈A_n and any k≥0,m_k(Z_n~2)≤m_k{A_n)≤m_k(Z_n~1) and i_k(Z_n~2)≥i_k(A_n)≥i_k(Z_n~1),with the left of equalities holding for all k only if A_n=Z_n~2;the right of equalities holding for all k only if A_n=Z_n~1,where Z_n~1 and Z_n~2 are the chain of type one and the chain of type two, respectively (see figure 6 (a) and 6 (b)).In chapter 4, we give the matching (coefficient) polynomial of the chain of type two Z_n~2 (see figure 6 (b)).In chapter 5, we discuss the extremal h-polygonal chains (h≥5) on Wiener numbers. Let W(A_n) be the Wiener number of A_n.In the fourth chapter, we show that for any polygonal chain A_n∈A_n,W(Z_n~2)≤W(A_n)≤W(Z_n~3),with the left of equalities holding for all k only if A_n=Z_n~2;the right of equalities holding for all k only if A_n=Z_n~3,where Z_n~2 and Z_n~3 are the chain of type two and the chain of type three, respectively (see figure 14 (b) and 14 (a)).