Graph energy is one of the most significant aspects in mathematical chemistry that has generated a lot of interest in researchers over a long period of time. The concept of energy of a vertex, introduced by Arizmendi et al., provides a local spectral perspective to the graph energy, highlighting the significance of individual vertex energies and their contribution to the total energy of the graph, and is therefore becomes important to understand the role of each vertex in understanding the spectral properties of the graph. In literature, various authors have worked on studying the properties of vertex energy and computing the same for various classes of graphs using the method explained in the original article. Particularly, computing the vertex energies of integral graphs is commonly found in recent studies as it involves graphs with only integer eigenvalues, thereby ensuring efficiently obtaining exact values of vertex energies and not numerical approximations. Recently, Gutman et al. have proposed a novel approach to compute the vertex energy of a graph using eigenvalues and eigenvectors of a graph with a specified number of vertices. In the present study, we adopt this innovative approach, particularly useful with graphs having too many distinct eigenvalues and large number of vertex symmetries, to determine the vertex energies for all 22 connected integral graphs of order eight by finding all the eigenvalues and their corresponding orthonormal eigenvectors using the Gram-Schmidt process.
Shrikanth et al. (Fri,) studied this question.