Borderenergeticity Under Graph Operations

In 1978,graph energy was defined by Gutman,an internationally famous theoretical chemist,as the sum of the absolute values of the eigenvalues of the adjacency matrix.If a graph with n vertices satisfies E(G)=2(n-1),the graph is called a borderenergetic graph.The borderenergetic graph,Laplacian borderenergetic graph,signless Laplacian borderenergetic graph,Laplacian-,and signless Laplacian-in this paper are simply written down as BE graph,LBE graph,QBE graph,L-,and Qrespectively.In 2015,Gong Shicai,Li Xueliang et al.constructed an non-complete BE graph by using tensor product,line graph,strongly regular graph and union and complement operation of graph.In 2016,Deng Bo,Li Xueliang et al.constructed regular non-complete BE graphs.In order to further understand the structural properties of BE graphs,this paper will study the structural properties of(Laplacian)BE graphs under some graph operations and the borderenergeticity of any connected graphs under some graph operations.The specific research contents are as follows:(1)The structural properties of BE graph under line graph operation are studied,and the bounds of line graph energy of graphs are considered under the condition of minimum degree;More generally,we study the structural properties of the line graph of any connected graph,that is,whether the line graph of any connected graph is of borderenergetic graph;The borderenergeticity of graphs under the action of cartesian product is studied;(2)Based on the structural properties of BE graph and its complement graph studied by Deng Bo and Li Xueliang in 2021,it is extended to LBE graph;Using the results of Nordhaus-Gaddum-type,the L-energy of the complement of LBE graph is studied;More generally,we study the L-borderenergeticity of the line graph of any connected graph;(3)Investigate the structural properties and some other results of QBE graph and its complement graph.