The Incidence Energy And Laplacian Incidence Energy Of Graphs

Algebraic graph theory is an important branch of graph theory, algebraic method is applied in the algebraic graph theory. In this thesis, we have studied the incidence energy and the Laplacian incidence energy of graphs.In1970s, Gutman defined the concept of the energy of a graph, the sum of all the absolute values of the eigenvalues of its adjacency matrix. By the further study of energy, the concept has generalized into other matrices of a graph, such as the incidence energy, Laplacian energy, Laplacian-energy like invariant, the normalized incidence energy, the skew energy of oriented graphs and so on.In this thesis, two topics have been studied. The first one is the study on the incidence energy of graphs, the sum of singular values of incidence matrix. We characterize the trees with the fifth smallest to seventh smallest incidence energy and the trees with the fourth greatest to the (?)(n+1)/4」-th largest incidence energy among all trees on n vertices. The second one is defining a new energy, Laplacian incidence energy, give sharp upper and lower bounds of it and show a close relation with other energies.In Chapter1, we introduce research background, some notations, the prop-erties and results about the energy of graphs. In Chapter2, we give a summary of research on incidence energy of graphs, and then prove the result on incidence energy of trees. In Chapter3, we first define the Laplacian incidence energy of graphs, and then have present its bounds and relations with others.