The Study On The Lower Bound Of Kekulé Count And Stability Of Fullerenes

Fullerene is another all-carbon crystal structure which is discovered after graphite and diamond.Fullerene and its derivatives have broad application in the fields of electrochemistry,anticancer drugs,superconducting materials and life sciences.This thesis studies the stability of fullerenes by the theory and calculation of fullerene graph.The Kekulé structure is a very important graph invariant.This paper introduces the3 k + 2 layer neighborhood of the vertex in the dual graph of fullerene.This method improves the lower bound of the Kekulé structure count.Calculate the number of 3k + 2layer neighborhood in the dual graph of fullerene,and color some vertices of the 3k + 2layer neighborhood.The 3k + 2 layer neighborhood which contains partially colored vertices is extended to the entire dual graph by the four-color theorem.Taking a five-layer neighborhood as an example,its lower bound of Kekulé count is better than the current optimal result.When k is sufficiently large,the lower bound given by this method doubles the current optimal result.There is an equality relationship between the permanent and Kekulé count.Based on the numerical computing results of the permanent and Kekulé count,it shows that the lower bound of kekulé count still has much room for improvement.The mapping relationship between the fullerene graph and even cycles is established in order to further study the lower bound of Kekulé count.It is an important research topic to predict the stability of fullerene structures by the graph invariants.The conclusions of the fullerene stability predictors are numerous but rather messy.The different predictors have their own valid ranges.They even contradict each other.A single graph invariant is difficult to form an effective stability predictor.Moreover,the stability of isomers lacks a unified standard for research.This paper analyzes and synthesizes the research results obtained from a large number of existing laboratory and theoretical chemistry calculations,and gives a standard for the structural stability of fullerenes.The existing graph invariant indicators related to the stability of fullerenes are collected comprehensively.Try to combine different graph invariants to form predictive power.The graph invariants with strong predictive power that are related to molecular structures and eigenvalues are selected.Based on the Clar number,a heuristic algorithm and an integer optimization model of aggregate sort are proposed.The results of heuristic algorithm have good agreement with the standard of fullerene structure stability.The characteristic and permanental polynomials are one of the most important research topics in chemical graph theory.They can characterize the molecular structure.Furthermore,they can be used as predictors of the stability of fullerene molecules.As the coefficient item increases,Sachs graph contains more structures which are more complex.It is more difficult to obtain coefficients of higher items.In this paper,several higher item coefficients and more relationships between coefficients are obtained.