Study On Strucural Properties Of Signless Laplacian Borderenergetic Graph

In 1978,Gutman,the international chemist,defined graph energy asε（G）that is,the energy of graph G isε（G）=_i=1ⁿ|λ_i|.Andλ_iis the eigenvalue of adjacent matrix of G.The graph energy of borderenergetic graph G isε（G）=|λ_i|=2（n-1）,where n is the order of G.Similarly,the m of Laplacian borderenergetic graph（L-borderenergetic graph）and signless Laplacian borderenergetic graph（Q-borderenergetic graph）have been proposed successively.In 2015,Gutman,Furtula,X.L.Li et.successively calculated the borderenergetic graph and its complement,L-borderenergetic graph and its complement,and obtained the corresponding structural properties by complementary graph operation.In order to further understand the energy and properties of the graph,this paper will perform graph operations on the Q-borderenergetic graph,and obtain the following specific contents of the structural properties of the Q-borderenergetic graph,the results are as follow:（1）The structure and properties of Q-borderenergetic graph under operation of complement graph are studied,and the related properties of the complement graph of Q-borderenergetic graph are obtained.At the same time,it is proved that under certain conditions of First Zagreb index,graph G and its complement graph （?） cannot be a Q-borderenergetic graph at the same time.This result reveals that the Q-borderenergetic graph,the borderenergetic graph and the L-borderenergetic graph have the same properties under operation of complement.（2）The structure properties of Q-borderenergetic graph under operation of line graph are studied,and the corresponding structure properties are obtained when there is a certain relationship between the number of edges and the order of a graph G,the graph G and its line graph L_G cannot be a Q-borderenergetic graph at the same time.On the other hand,the related structural properties of the line graph of the Q-borderenergetic graph are obtained.This result is the first time to study the complement of Q-borderenergetic energy graph and its line graph,and to describe the structure properties of Q-borderenergetic graph and its line graph more profoundly,providing a new way to construct new Q-boundary energy graph.（3）Using the degree sequence of graphs,we obtain some new criteria for Q-borderenergetic graph with these criteria.For most graphs,we can intuitively and effectively judge Whether the graph G is a Q-borderenergetic graph.