Vertex energy distributions in regular graph structures

The energy of a vertex $v_i$ in a graph $G$ is defined as ${\mathcal{E}}_G(v_i)=\mathrm{\mid }A{\mathrm{\mid }}_{ii}$, where $A$ is the adjacency matrix of $G$, $A^{\ast }$ denotes the conjugate transpose of $A$, and $\mathrm{\mid }A\mathrm{\mid }=(A{A}^{\ast }{)}^{1\mathrm{/}2}$. The total energy of the graph, $\mathcal{E}(G)$, is then the sum of the energies of all vertices: $\mathcal{E}(G)={\mathcal{E}}_G(v_1)+{\mathcal{E}}_G(v_2)+\cdots +{\mathcal{E}}_G(v_n)$. 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.