Eccentricity Laplacian energy of a graph

A BSTRACT Let G be a simple, finite, undirected and connected graph. The eccentricity of a vertex v is the maximum distance from v to all other vertices of G . The eccentricity Laplacian matrix of G with n vertices is a square matrix of order n , whose elements are el ij , where el ij is -1 if the corresponding vertices are adjacent, el ii is the eccentricity of v i for 1 <= i <= n , and el ij is 0 otherwise. If e 1 , E2,... , e n are the eigenvalues of the eccentricity Laplacian matrix, then the eccentricity Laplacian energy of G is ELE(G) = X n i=1 |Ei- avec(G)|, where avec(G) is the average eccentricities of all the vertices of G . In this study, some properties of the eccentricity Laplacian energy are obtained and comparison between thge eccentricity Laplacian energy and the total pi -electron energy is obtained.