The Estimation Of The Energy Of The Digraphs And The Construction Of The Equienergetic Digraphs

As a generalization of graph energy,the digraph energy is widely used in computer networks,biology,chemistry and other fields.Let D be a digraph,A（D）be its adjacency matrix,z₁,z₂,...,zn be the eigenvalues of A（D）,then the energy of the digraph D is expressed as the sum of the absolute values of the real parts of the eigenvalues of its adjacency matrix,that is,E（D）=（?）|Rez_i|.In 2015,Gutman et al.proposed the concept of the borderenergetic graph.If the energy E（G）of a graph G of order n is E（G）=2（n-1）,then the graph G is called borderenergetic graph,where n is the number of vertices of graph G.Similarly,the borderenergetic digraph can be defined.For the direction of the arc in digraphs and its structure is complex,it is difficult to study the structure and properties of the borderenergetic digraph.As a generalization of the general graph,the study of the borderenergetic digraph can promote the further development of graph energy problem.In addition,based on the symmetric of digraphs,the Nikiforov energy of the digraph is also the energy of the digraph.In this paper,the Nikiforov energy of the digraph is also studied.In this paper,we use mathematical induction,inequality scaling and graph construction methods to obtain the following results around the extreme properties of the energy of the digraphs and the borderenergetic digraph:（1）For the strongly connected regular digraph,first,the spectrum of its complement graph is obtained.Secondly,by using the join graph operation,the spectrum of the join graph of two strongly connected regular digraphs is obtained.Finally,a method of constructing a class of digraphs is proposed to obtain a new class of the borderenergetic digraph.Furthermore,this paper explores the relationship between the energy of strongly connected regular digraphs and their complement graphs.（2）According to the definition of the energy of the digraph,to start with,the relationship between the real part and the imaginary part of the eigenvalues of the digraph is limited,and then the lower bound of the energy of the digraph related to the trace of the matrix is obtained by using the inequality scaling.Secondly,the characteristic polynomial of the digraph is obtained by using Sachs theorem,and then the spectrum of the digraph is obtained according to the condition of taking the equal sign.Finally,we construct a class of digraphs whose energy reaches the lower bound.This result studies the case of directed 4-cycles,and generalizes the result of Rada.（3）According to the definition of the arc energy of a digraph,the bounds of the arc energy on the outdegree,the indegree and the maximum degree are obtained by using the Cauchy Schwarz inequality.Using the relationship between the arc energy and the Nikiforov energy of the digraph,the bounds of the Nikiforov energy of the digraph on the number of arcs and the maximum degree are obtained.This result reveals the close relationship between the Nikiforov energy and the maximum degree from the angle of the arc energy of the digraph.