Studies On Anti-Forcing Spectrum Of Some Graphs

Let G be a simple,connected,finite graph,M a perfect matching(or Kekule structure)of G.The anti-forcing number of M of G is the minimal number of edges not in M whose removal makes M a unique perfect matching of the resulting graph.The set of anti-forcing numbers of all perfect matchings of G is called the anti-forcing spectrum of G,the smallest number of the anti-forcing spectrum of G is called the anti-forcing number or minimum anti-forcing number of G,the largest number is called the maximum anti-forcing number of G.The anti-forcing spectrum is obtained by adding the multiplicity of each anti-forcing number on the basis of the anti-forcing spectrum.This paper first calculates the refined anti-forcing spectrum and its continuity of ladder graph.By classifying and counting all perfect matchings of ladder graph according to their anti-forcing numbers,we get a formula about Fibonacci numbers.Then,based on the number of anti-forcing ladder graph,calculate the anti-forcing numbers of circular ladder graph,Mobius ladder graph and incomplete ladder graph ILn-3 which by added or deleted edges from the ladder graph,and get the refined anti-forcing spectrum and their continuity of circular ladder graph,Mobius ladder graph and incomplete ladder graph ILn-3.By classifying and counting all perfect matchings of circular ladder graph,Mobius ladder graph and incomplete ladder graph ILn-3 according to their anti-forcing numbers,we get a formula about Lucas numbers.