On Vertex Eccentricity Labeled Energy of A Graph

In this paper, we introduce a new spectral graph invariant called the Vertex Eccentricity Labeled Energy (VELE). VELE is derived from the ‎eigenvalues of the Vertex Eccentricity Labeled (VEL) matrix, whose off-diagonal entries for a connected graph G are defined as the sum of ‎eccentricities of vertex pairs, and zero otherwise. We explore fundamental algebraic and spectral properties of VELE, including its trace, ‎characteristic polynomial, and energy bounds. Closed-form formulas for VELE are provided for several classical graph families, such as ‎complete graphs, cycles, wheels, and stars. Additionally, the behavior of VELE is studied under common graph operations such as Cartesian product, splitting, and m-splitting, highlighting distinctive spectral patterns related to vertex eccentricity. These results expand the family ‎of spectral descriptors and offer new analytical tools in structural graph analysis‎.