Authors:
Yang Yang, Yanyan Song, Zhanjun Si, Haiyan Qiao
Abstract
Consider a simple undirected connected graph \( G \) that has an adjacency matrix \( \textbf{\textit{A}} \). For a vertex \( i \in
V(G) \), the vertex energy (VE) of \( i \) in \( G \) is \( E_{\pi}(i)=|\mathbf{A}|_{ii} \), where
\( |\mathbf{A}|=\left(\mathbf{AA}^*\right)^{1 / 2} \). Furthermore, the graph energy of \( G \) is \( E_{\pi}(G)
= \sum_{i=1}^n |\lambda_i| = \sum_{i=1}^nE_{\pi}(i) \), where \( \lambda_1,\lambda_2,\ldots,\lambda_n \) 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.
