The2-Resenance Of Fullerene Graphs

Let G be a plane graph or a2-cell embedding on some surface. A set H of disjoint even faces of G is a resonant pattern (or sextet pattern if all faces in H are hexagons) if G-H has a perfect matching (or Kekule structure in chemistry). G is k-resonant if any i (i≤k) disjoint even faces form a resonant pattern. The in-spiration of k-resonant structure comes from Clar’s aromatic sextet theory and the conjugated circuit theory of Randic’s. The k-resonance was firstly proposed for ben-zenoid systems. Experimentally speaking, for two isomeric benzenoid hydrocarbons, the one with the maximum number of mutually resonant sextets among all Kekule structures should be more stable. Then the research of k-resonance was generalized to plane graphs and2-cell embeddings on surfaces, such as coronoid systems, open-ended carbon nanotubes, spherical fullerene graphs, toroidal fullerene graphs, Klein-bottle fullerene graphs, boron-nitrogen fullerene graphs, polygonal systems, cubic bipartite polyhedral graphs.Fullerene graphs are3-connected3-regular plane graph having only pentagonal faces and hexagonal faces. Such graphs are suitable models for fullerene molecules. In2009, Ye and Zhang showed that every fullerene graph is1-resonant, and the k-resonant (k>3) fullerene graphs are only the nine graphs F20,F24,F28,F32,F361,F362,F40, F48and F60, but not all fullerene graphs are2-resonant. They also proved each leapfrog fullerene graph is2-resonant and asked a problem:whether every fullerene graph without adjacent pentagons (and thus such graphs are called IPR fullerene graphs) is2-resonant. In2011, Kaiser et al. gave a positive answer to the problem. But the2-resonance of fullerene graphs has not been resolved.For k≥3an integer, a (k,6)-fullerene is a planar cubic graph whose faces are only k-gons and hexagons. The only values of k for which (k,6)-fullerene exists are3,4and5. A (4,6)-fullerene is a boron-nitrogen fullerene molecular graph and a (5,6)-fullerene is the ordinary carbon fullerene molecular graph. Here is a natural motivation for us to consider the k-resonance of (3,6)-fullerene graphs. Around these two issues, in this thesis, we study the k-resonance of (3,6)-fullerene graphs and the2-resonance of fullerene graphs.There are five chapters in this thesis. In Chapter one, we first introduce some basic concepts, terminologies and notations. Then we introduce the background and research progress of resonance of graphs. At last, we outline the main results obtained in the following chapters.In Chapter two, we consider the hexagonal resonance of (3,6)-fullerene graphs. For a (3,6)-fullerene graph G, we can know its connectivity is either2or3. The structure of a (3,6)-fullerene graph with connectivity3is well know. So in this chapter we firstly give a structural theorem about the (3,6)-fullerene graph with connectivity2. Then we distinguish two cases to prove that a (3,6)-fullerene graph G with connectivity2is not1-resonant and a (3,6)-fullerene graph G with connectivity3is1-resonant (except for one graph) but not2-resonant (except for two graphs).In Chapter three, we consider the2-resonance of fullerene graphs without sub-graph L or R. Since the IPR fullerene graphs don’t include the subgraph L or R, inspired by the proof of Kaiser’s, we generalize the result of Kaiser’s (every IPR fullerene graph is2-resonant) to the fullerene graphs containing no the two subgraphs, and show that a fullerene graph without subgraph L or R is2-resonant except for eleven fullerene graphs.In Chapter four, we consider the2-resonance of extremal fullerene graphs and the2-resonance of fullerene graphs with at most60vertices. By means of the charac-terizations about the extremal fullerene graphs with vertices no less than60, together with the result of chapter three, we firstly testify that every extremal fullerene graph with no less than60vertices is2-resonant. Then we find all non2-resonant fullerene graphs with at most60vertices by computer.In Chapter five, we give a description about the2-resonance of fullerene graphs, that is, a fullerene graph is2-resonant if and only if it is not isomorphic to one of the119fullerene graphs or it does not possess one of the47graphs as subgraph.