The Sachs theorem and its application on extended adjacency matrix of graphs

Based on the extended adjacency matrix, the spectral radius and the energy of the extended adjacency matrix are found that they possess high discriminating power and correlate well with a number of physicochemical properties and biological activities of organic compounds. In this research, by establishing a relationship between directed graphs and undirected graphs, we mainly present the Sachs theorem and the Coulson’s integral formula of the energy of the extended adjacency matrix of G . By using the formula, we can compare the energies of the extended adjacency matrices of two graphs and we find that the graph energy is less than the extended energy in the case of a tree.