Local vertex energy of brooms and double-star trees

Source abstract

Abstract In this paper, we investigate the local vertex energy of trees, a spectral invariant that quantifies the contribution of individual vertices to the total energy of a graph. Inspired by questions posed in recent work on local energy, we investigate this quantity in two structured families: broom trees and double-star trees. For broom trees, we identify the vertices that attain maximal and minimal local energy, both within a fixed broom and across the class of all brooms with a given number of vertices. While the general case of double-broom trees remains open due to the complex behavior of vertices along the central path, we provide a complete solution for the subclass of double-star trees. These results enhance the overall understanding of vertex-level energy distribution and encourage further developments in the spectral analysis of trees.