The Number Of The Perfect Matchings In N-Prism And The Face Resonance Of Two Types Of Graphs

Matching theory is a basic branch of graph theory.In chemistry,the perfect matching of the benzenoid system is also called the Kekule structure.Clar proved that the construction of an appropriate aromatic sextet through the Kekule structure of hydrocarbons can predict various electronic properties of this hydrocarbon.As a very important topological index,the perfect matching number has been applied in many fields.In benzenoid system,there is a concept closely related to perfect matching,that is,resonance.The concept of resonance comes from Clar’s aromatic sextet theory and Randic’s conjugated circuit model.According to Clar’s aromatic sextet theory,Clar found that aromatic hydrocarbon molecules with more mutually resonant hexagons are more stable.Resonance also plays an important role in predicting molecular stability and resonance energy.Based on the above facts,it is of great significance to study the perfect matching number and resonance problems of some graphs in this paper.Firstly,we study a special class of cartesian product graph n-prism,divide the perfect matchings of n-prism,then by means of the formula of Fibonacci sequence,thus obtaining the formula of the number of perfect matchings in n-prism;Finally,the resonance of n-prism is discussed,and it is obtained that n-prism is k-resonant(k≥1)and maximally resonant.Secondly,we study the face resonance of the {(3,4),4}-fullerene graph S,and show that every quadrangular face in S is resonant.Here a {(3,4),4}-fullerene graph is a 4-regular planar graph whose faces are of length 3 or 4.Finally,we study the structural properties of the {(2,3),6}-fullerene graph F,where{(2,3),6}-fullerene graphs are 6-regular planar graphs whose faces are only 2-length and 3-length.This paper proves that the cyclical edge-connectivity of {(2,3),6}-fullerenes is 4 or 6 or 8,and characterizes the structures of {(2,3),6}-fullerenes with cyclical edgeconnectivities 4 and 6,respectively.Furthermore,we show that every 2-length face of a{(2,3),6}-fullerene is resonant.