Abstract
Let \( G \) be a graph with adjacency matrix \( A \). The energy of \( G \) is denoted by \( \mathcal{E}(G) \) and defined as the sum of the absolute values of the eigenvalues of \( A \). In this paper we study the problem of variation of the energy when a vertex is deleted. Concretely, we show that if \( G \) is a graph and \( G^{\left(j \right) } \) is the graph obtained from \( G \) by deleting the vertex \( v_{j} \) of \( G \), then
\[
\mathcal{E}(G) - \mathcal{E}(G^{\left(j \right) }) \leq 2\sqrt{d_{j}},
\]
where \( d_{j} \) is the degree of \( v_{j} \). Moreover, equality occurs if and only if the connected component of \( G \) containing \( v_{j} \) is isomorphic to a star tree and \( v_{j} \) is its center. Afterwards, we introduce a new approach to the local energy of a vertex and initiate the study of its basic properties.
