Vertex energy distributions in regular graph structures

The energy of a vertex vᵢ in a graph G is defined as EG(vᵢ) = |A|ᵢᵢ, where A is the adjacency matrix of G, A^* denotes the conjugate transpose of A, and |A| = (AA^*)1/2. The total energy of the graph, E(G), is then the sum of the energies of all vertices: E(G) = EG(v₁) + EG(v₂) + + EG(vₙ). In this paper, we compute the vertex energy for several well-known regular graphs, including the Frucht graph, Desargues graph, Tutte-Coxeter graph, Heawood graph, Shrikhande graph, and Petersen graph.