Consider a simple undirected connected graph $G$ that has an adjacency matrix $\mathbf{A}$. For a vertex $i\in V(G)$, the vertex energy (VE) of $i$ in $G$ is $E_{\pi }(i)=|\mathbf{A}{|}_{ii}$, where $|\mathbf{A}|={\left({\mathbf{AA}}^{\ast }\right)}^{1/2}$. Furthermore, the graph energy of $G$ is $E_{\pi }(G)=\sum _{i=1}^n|{\lambda }_i|=\sum _{i=1}^n{E}_{\pi }(i)$, where ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ are the eigenvalues of $\mathbf{A}$. This paper introduces new computational equations for the vertex energy of graphs based on an equitable partition strategy, star sets, and the Estrada-Benzi approach. Furthermore, this paper provides the VE bounds of the graphs using a multi-digraph that corresponds to the quotient graphs of $G$. Additionally, this study calculates the VE upper bounds of the vertex's maximum degree for the wheel, the friendship, and endohedral fullerenes graphs more accurately.