On Connectivity And Resonance Of (5,6,7)-Fullerene Graphs

(5,6,7)-fullerene graphs are plane graphs whose faces are only pentagons,hexagons and heptagons.For a(5,6,7)-fullerene graph,a 6-long face cycle inis said to be resonant,if its boundary(or itself)is an-alternating cycle for some perfect matchingof.A ring chain of a(5,6,7)-fullerene graph is a subgraph formed by a circular arrangement of9)faces such that only two consecutive faces have edges in common and these common edges are is independent,denoted as[(a₁,(a₂,...,(a₉)],where(a_i)?{5,6,7}and4)=1,2,...,9)is defined as the length of the corresponding face cycle.This paper mainly studies the connectivity,cyclically edge connectivity,and res-onance of(5,6,7)-fullerene graphs.The first chapter mainly introduces the definitions and symbols involved in this paper,as well as the research significance and progress of(5,6,7)-fullerene graphs.The second chapter shows the connectivity of(5,6,7)-fullerene graphs is 2 or 3 and cyclically edge connectivity is 2,3,4,or 5.Two characterizations are given:The connectivity of a(5,6,7)-fullerene graph is 2 if and only if there are[7,7]or[6,7]-ring chains;The connectivity of a(5,6,7)-fullerene graph is 3 if and only if there are[7,7,7],[6,7,7],[5,7,7]or[6,6,7]-ring chains.Chapter 3 discusses the1-resonance of(5,6,7)-fullerenes,first indicating each 6-long face cycle in the(5,6,7)-fullerene graph of the 4-and 5-cyclically edge connectivity are all resonant.Secondly,for(5,6,7)-fullerene graphs with the connectivity of 3 and no nontrivial cyclically edge cuts of size greater than 5,we characterize 24 types of subgraphs with 6-long face cycle of non-resonant.Finally,for(5,6,7)-fullerene graphs with the connectivity of 2 and no nontrivial cyclically edge cuts of size greater than 5,we characterize the subgraph types of non-resonant 6-long face cycle.The fourth chapter summarizes the results obtained in this paper and points out the direction and content for further research.