Locally Equienergetic Graphs

For a given graph $G$, let $G^{(j)}$ denote the graph obtained by the deletion of vertex $v_j$ from $G$. The difference $\mathcal{E}(G)-\mathcal{E}(G^{(j)})$ quantifies the change in the energy of $G$ upon the removal of $v_j$, termed as the local energy of $G$ at vertex $v_j$, as defined by Espinal and Rada in 2024. The local energy of $G$ at vertex $v$ is denoted by ${\mathcal{E}}_G(v)$. The local energy of the graph $G$, therefore, is the summation of these vertex-specific local energies across all vertices in $V(G)$, expressed by $e(G)=\sum {\mathcal{E}}_G(v)$. Two graphs of the same order are defined as locally equienergetic if they have identical local energy. In this paper, we have investigated several pairs of locally equienergetic graphs.