The Clar Number Of Fullerene Graphs On Surfaces And The Related Extremal Problems

A fullerene graph is a finite trivalent graph on the sphere with only hexagonal and pentagonal faces. It is the molecular graph of a novel spherical carbon found by Kroto et al. in 1985. As an important index, the Clar number can be used to measure the stability of the hydrocarbons and fullerenes.Zhang and Ye proved that the Clar number of a fullerene graph of order n is bounded above by [n/6] - 2. Then they gave a characterization of fullerene graphs with at least 60 vertices attaining the maximum Clar number n/6-2. Based on such a characterization, they constructed all 18 fullerene graphs whose Clar numbers attain the maximum value 8 among all fullerene isomers of C60. In this thesis we find that the bound [n/6] - 2 can be improved to [n/6] - 3 for fullerene graphs whose order is congruent to 2 modulo 6. Amongst all 11 experimentally characterised fullerenes, there are 9 fullerenes, whose molecular graphs attain these two bounds, that is, C60:1(Ih), C70:1(D5h), C76:1(D2), C78:1(D3), C82:3(C2), C84:22(D2), 84:23(D2d), C80:1(D5d) and C80:2(D2).In 2000, Deza et al. considered the fullerene graphs on other surfaces. Fullerene graphs only exists on the sphere, projective plane, torus and Klein bottle. In this thesis, we mainly study the upper bounds of the Clar number of fullerene graphs and the characterization of the extremal classes.This thesis contains five chapters. In Chapter 1, firstly, we introduce some basic definitions and notations of graph theory. Then the definitions of fullerene graphs and some basic properties are presented. Finally, we list the main results of this thesis.In Chapter 2, as a generalization of spherical fullerene graphs we obtain a sharp upper bound for the Clar number of any fullerene graph F with n vertices on surface Σ, that is, c(F)≤[n/6]-χ(Σ), where χ(Σ) stands for the Euler characteristic of E. Moreover, we present five families of projective fullerene graphs attaining the upper bound n/6-1, and characterize all toroidal and Klein-bottle fullerene graphs whose Clar numbers are n/6.In Chapter 3, we mainly characterize the fullerene graphs whose Clar numbers attain the bound n/6 - χ(Σ). Firstly, we show that the Clar number of a (4,6)-fullerene graph on surface Σ is bounded above by n/6+χ(Σ), and characterize the (4,6)-fullerene graphs on surface Σ attaining this bound in terms of perfect Clar structure. Then we give some equivalent characterizations of fullerene graphs on torus and Klein bottle with Clar numbers n/6 which extend the results in Chapter 2. Then, by introducing the concept of the diagonalization of (4,6)-fullerenes on the sphere and projective plane, we establish a relation between the extremal fullerene graphs and the extremal (4,6)-fullerene graphs on the sphere and projective plane, and characterize the fullerene graphs on the sphere and projective plane whose Clar numbers attain n/6 - χ(E). Moreover, we also discuss the relation between the projective fullerene graphs whose Clar numbers are n/6 - 1 and the spherical fullerene graphs whose Clar numbers are n/6 - 2.In Chapter 4, we find and prove that the Clar number for a spherical fullerene graph, whose order is congruent to 2 modulo 6, is bounded above by [n/6] -3. Further, we give a characterization of spherical fullerene graphs, whose order is congruent to 2 (respectively,4) modulo 6, attaining maximum Clar number [n/6] - 3 (respectively, [n/6]-2). Finally, we show that for each even number n ≥ 20 of vertices with the exceptions of n = 22 and n = 30 there is at least one spherical fullerene graph satisfying: if its order n is congruent to 0 or 4 modulo 6, then its Clar number is [n/6]-2; otherwise, [n/6]-3.In Chapter 5, we study the Clar number of the projective fullerene graphs whose order is congruent to 2 or 4 modulo 6. We characterize the projective fullerene graphs, whose order n is congruent to 2 or 4 modulo 6, attaining maximum Clar number [n/6]-1.