Construction Of Some Borderenergetic Graphs And Research On The Properties Of Graphs

Based on the eigenvalues of graph,Graph energy(denoted by ε(G))was proposed by Professor Gutman in 1977.Graph energy is significant invariant for describe the structural stability of graphs.It is extensively studied in mathematics,theoretical chemistry and computer science and so on.If the energy of graph G(on n vertices)is:ε(G)=2(n-1),then it is naned as borderenergetic graph(It is abbreviated as BEG).Since the(Signless)Laplacian energy is a generalization of graph energy based on matrix,we can define(Signless)Laplacian BEG similarly.It is shown that the structures of most BEGs are complex,and many of their corresponding structural properties are not clear.Therefore,it is of vital significance to depict the properties of structure of BEGs or to study their topological indices(numerical representation of graph structure).In 2012,Gutman and Wagner extended the graph energy from another perspective.They defined the matching energy based on the eigenvalues of matching polynomials,and obtained a relational expression among graph energy,topological resonance energy and matching energy.It is claimed that the application background of chemistry of matching energy can be traced back to the 1970s.This paper chiefly focuses on the BEGs and the matching energy of the graph and the particular study contents are as below:(1)Based on the regular graph,the Q-spectrum of its complement is obtained;(2)According to the characteristics of join graph,the spectrum of join graph of two special structure graphs is obtained,and a series of Signless Laplacian BEGs Gs,s≥0 are constructed by using the method of continuous join graph for special structure graphs and the spectrum of j oin graph;(3)Some bounds on the order of Signless Laplacian BEGs are obtained by using the graph invariants such as the number of edges;(4)Based on the energy graph classes of(Signless)Laplacian BEG constructed by Bo Deng and Yaoping Hou,a more general graph class is constructed,and sufficient conditions are obtained to determine whether it is L-BEG(or Q-BEG);(5)The relations among graph energy,Laplacian energy and Signless Laplacian energy are studied.It is proved that the three kinds of energy are equal when the graph is regular.A bound of the order of regular boundary energy graph is given;(6)A lower bound of matching energy of random bipartite graph is given,and it is generalized on random multipartite graph.