The Laplacian Energy Of The Special Graph

The famous Seven Bridge problem is the origin of the graph theory,and then it becomes an important branch of the Applied Mathematics.Laplacian energy problem of special graphs has been a hotspot issue in the field of graph theory.In 2006,concept of the Laplacian energy was first introduced by Ivan Gutman and Bo Zhou.It has not only theoretical value,but also has important practical significance,widely applied to chemical field,image segmentation hierarchy,the emerging field of computer science,and so on.Let G =(V,E)be a non trivial connected graph,edge set is E(G),the edge number is m,vertex set is V(G),the vertex number is n.Let the adjacency matrix is A(G),the matrix of points is D(G),it is the graph Laplacian matrix L(G)= D(G)-A(G).The Laplacian matrix has eigenvalues,therefore,?i(i = 1,2,3...n)can represent of the Laplacian eigenvalues.And ?1? ?2>...?n 0.The formula of Laplacian energy is LE(G)= ?i=1 n |?i-2m/n|In this paper,we focus on the mainly applied mainly applied of the problem of the energy of Laplacian.The main results and structure are as follows:(1)The basic concepts about the background of the Laplacian energy of graph and related conclusions are introduced;based on the method of the matrix eigenvalue calculation,thorn graph of complete graph of the lower and upper bound of Laplacian energy are obtained.(2)Based on the mathematical induction method and the classification discussion methods,the combined graph of circle graph,and the windmill graph of Laplacian energy expression and upper and lower bounds are obtained.(3)Based on the mathematical induction method,classification discussion methods,and idea of combined optimization,the Laplacian energy and Laplacian characteristic value of k-tree are calculated.