Different Graph Energies and Their Relation

This paper discusses one graph spectral-based invariant, graph energy.First, we define the adjacency, Laplacian, and signless Laplacian matrices associated with the graph.Then, the characteristic equations of these matrices are described.The graph spectra are obtained by solving these adjacency, Laplacian, and signless Laplacian characteristic equations.The roots of these characteristic equations are called eigenvalues of the respective matrices of the graph.The set of eigenvalues of the graph with their multiplicities is known as the spectrum.Subsequently, adjacency, Laplacian, and signless Laplacian spectra are obtained.The graph's energy is the sum of the absolute values of the eigenvalues of the matrix of the respective graphs.Consequently, energy, Laplacian energy, and signless Laplacian energy are obtained and established in their relation in this paper.