Research On Some Perfect Matching Properties Of (4,5,6)-fullerene Graphs

Fullerenes are the third allotrope of elemental carbon in a spherical,elliptic,or tubular form.It have had a profound impact on chemistry,physics,materials and medicine.The molecular graph of a fullerene is a plane（or spherical）cubic graph in structures whose faces are pentagons and hexagons,which is also called a（5,6）-fullerene graph.However,（4,6）-fullerene graphs is a plane 3-connected cubic graph,all of whose faces are quadrilaterals and hexagons.Several theoret-ical studies demonstrated that non-classical fullerenes with four-membered rings cannot be dismissed in advance.In fact,they may actually be stabilized by in-corporation of two four-membered rings through experiments.The presence of four-membered rings greatly enriches the world of fullerenes.（4,5,6）-fullerene graphs are plane（or spherical）cubic graphs whose faces are only quadrilaterals,pentagons and hexagons,which include all（4,6）-and（5,6）-fullerene graphs.Let p₄and p₅be the number of quadrilaterals and pentagons of a（4,5,6）-fullerene graph.Then,2p₄+p₅=12.The structural properties and isomer stabilities of（5,6）-and（4,6）-fullerene graphs were extensively investigated from both chemical and mathematical points of view.However,to our knowledge,a systematic study on non-classical fullerene graphs,i.e.,（4,5,6）-fullerene graphs,has not been found in mathematics.In this thesis,we study some perfect matching properties of（4,5,6）-fullerene graphs from the view of mathematical.In this thesis,we mainly study four properties of（4,5,6）-fullerene graphs.Since the stability of molecules has always been a fundamental concern of chemists and“resonance theory”is one of the most important indicators of molecular stability,we firstly consider the resonance of（4,5,6）-fullerenes and completely characterize the（4,5,6）-fullerenes whose every even face is resonant;we decide the（4,5,6）-fullerene graphs with anti-Kekulénumber 3 and characterize all 2-extendable（4,5,6）-fullerene graphs by the method of prohibited subgraphs;we finally study the anti-forcing number of（4,5,6）-fullerene graphs and characterize all（4,5,6）-fullerene graphs with the minimum anti-forcing number 3.This thesis contains five chapters as follows.In chapter 1,first we introduce some useful concepts,terminologies and no-tations.Then we introduce the research background and development of（4,5,6）-fullerene graphs.Finally,we list the main results of this thesis.In chapter 2,we mainly characterize the（4,5,6）-fullerene graphs each even face of which is resonant.For a（4,5,6）-fullerene graph F,an even face（or a cycle）is called resonant if its boundary（or itself）is an M-alternating cycle（i.e.,the edges of the cycle alternate in M and E（F）\M）for some perfect matching M of F.In this chapter,we first get the properties that the connectivity of any（4,5,6）-fullerene graph is 3 and every（4,5,6）-fullerene graph with at least one quadrilateral and one pentagon has cyclical edge-connectivity 4.Then,mainly based on the two properties,we mainly show that every quadrilateral face of a（4,5,6）-fullerene graph is resonant and all hexagonal faces are resonant except for three classes of（4,5,6）-fullerene graphs which are characterized as nanotubes with p₄=3 and p₅=6.Further,we show that all the resonant 6-cycles in（4,5,6）-fullerenes are just formed from all hexagonal faces except for one hexagon in the mentioned-above three types of nanotubes,and from all pairs of quadrilaterals with a common edge.In chapter 3,we study the anti-Kekulénumber of（4,5,6）-fullerene graphs and class the（4,5,6）-fullerenes whose anti-Kekulénumber are 3 and 4 by means of subgraphs.The anti-Kekulénumber of a connected graph G is the smallest cardinality of an edge set S of G whose deletion leads to G-S being connected and having no perfect matching.In this chapter,we determine all（4,5,6）-fullerenes with the anti-Kekulénumber 3,which consist of four sporadic（4,5,6）-fullerenes(F₁₂,F₁₄,F₁₈and F₂₀)and three classes of（4,5,6）-fullerenes with 2≤p₅≤6.Finally,we can determine whether a（4,5,6）-fullerene has the anti-Kekulénumber3 or 4 in a linear time algorithm and there is a（4,5,6）-fullerene with n vertices having the anti-Kekulénumber 3 for any even n≥10.In chapter 4,we study the 2-extendability of（4,5,6）-fullerene graphs.A con-nected graph G with a perfect matching and at least 2k+2 vertices is called k-extendable if any matching of size k is contained in a perfect matching of G.We know that every（4,5,6）-fullerene is 1-extendable and at most 2-extendable.In this chapter,we characterize all（4,5,6）-fullerenes with 2-extendability by us-ing the method of prohibited subgraphs.Thus,the extendability of（4,5,6）-fullerenes graphs is completely solved.What’s more,those non-2-extendable（4,5,6）-fullerenes consist of four sporadic（4,5,6）-fullerenes(F₁₂,F₁₄,F₁₈and F₂₀)and five classes of（4,5,6）-fullerenes.Further,we find that all（4,5,6）-fullerenes with the anti-Kekulénumber 3 are non-2-extendable.Naturally,for any even n≥10,there exists a non-2-extendable（4,5,6）-fullerene with n vertices.In chapter 5,we mainly characterize the（4,5,6）-fullerene graphs with the anti-forcing number 3.The anti-forcing number of a graph G is the smallest cardinality of an edge set whose deletion leads to the remaining graph having an unique perfect matching.In this chapter,we get that any（4,5,6）-fullerene graph has the anti-forcing number at least 3 and also characterize all（4,5,6）-fullerenes with the anti-forcing number 3 by extending subgraphs.Moreover,we find there always exists a（4,5,6）-fullerene graph with anti-forcig number 3 for every even n≥8.Finally,we describe the（4,5,6）-fullerenes with anti-forcing number 3 and p₄=1,2,3,5 respectively.