On The Anti-kekulé Number Problem Of Regular Graphs

In chemical graph theory,perfect matching of the graph is also called the Kekule structure.Because the carbon compounds without the Kekule structure are unstable,the study of the Kekule structure is very important.Let graph G be a connected graph with a perfect matching,and F is a subset of edges of graph G.If there is no perfect matching and connectivity in G-F then F is called an anti-Kekule set of G.The size of the anti-Kekule set containing the least elements in all anti-Kekulee sets of G is called the anti-Kekule number,which is denoted as ak(G).Anti-Kekule number of many special graph have been obtained,such as in-finite triangular,rectangular,hexagonal grids,fullerene graphs and general 3-regular graphs,etc.The vertex degrees of these graphs are small.This papaer mainly studies the anti-Kekule number problem for general regular graphs,which is divided into three chapters.In chapter 1,we first give an introduction to the basic concepts,notations and terms related to the graph theory used in this paper.Then the research background and progress of the anti-Kekule problem are introduced.Finally,summarize the results of this paper.In chapter 2,we obtain that the upper bound of the anti-Kekule number of all r-regular graphs(r≥3)is 2r-2 and is tight.Then for the r-regular graph with(r-1)-edge-connected we obtain that the lower bound of the anti-Kekule number is r,which is also tight.In chapter 3,we study the anti-Kekule number problem of regular graphs un-der certain conditions.First,we obtain the sufficient and necessary condition that ak(G)= r under the condition of non-trivial odd cut and non-bipart it e graph.Then the lower bound of anti-Kekule number of the bipartite r-regular graph is r+1.Finally,we considered the anti-Kekule number of the regular graph with cut-vertex.