The K-resonance Of （4,5,6）-fullerene Graphs

Fullerenes are the third allotrope of elemental carbon,which exist in spherical、elliptic or tubular structures.It has high application value in physics,chemistry and biomedicine,so it is very important to study it.The molecular graph of a fullerene is a plane（or sphere）3-regular graph containing only pentagonal and hexagonal faces,so it is called（5,6）-fullerene graph.The concept of 6)-resonance was put forward by Zheng Maolin in studying hexagonal systems.For the fullerene graph ,the 6)-resonance is defined as follows: the set of some even facial cycles in is said to be a resonance set,if there exists a perfect matching such that every even face rings in are -alternating cycle.In particular,if any 4)（0≤ 4）≤6))even facial cycles in the can be composed of a set of resonance,called graph as 6)-resonance.For（4,6）-fullerene graphs and（5,6）-fullerene graphs,the 6)-resonance has been resolved.Currently Zhao Lifang proved that（4,5,6）-fullerenes graphs in addition to the three classes of graph is satisfied 1-resonance,and determine which（4,5,6）-fullerenes graph is 6)-resonance（6）≥ 2)has not been solved.In this thesis,the 2-resonance of（4,5,6）-fullerenes graphs and 3-resonance of（4,5,6）-fullerenes graphs are studied.We use Tutte’s theorem to exclude some structures which do not satisfy 2-resonance.For 3-resonance,the （）and （）are classified according to the maximum number of quadrilateral and pentagonal faces in（4,5,6）-fullerene.In addition,all（4,5,6）-fullerenes satisfying 3-resonance are characterized by patch extension,and the 3-resonance is also maximal resonant.