Juan Rada

Vertex–degree–based (VDB) topological indices assign to each edge a weight depending only on the degrees of its endpoints. We develop a unified linear and geometric framework for studying these indices on graph classes with bounded maximum degree. By encoding each graph through its edge–type vector, every VDB index becomes a linear functional on a finite dimensional space. The analytic form of the generating function plays no essential role: a VDB index is completely determined by its values on degree pairs, and we show that every such index coincides with one induced by a symmetric polynomial. An explicit construction is given for the Sombor index. We then study discrimination. Strong discrimination is infeasible for VDB indices on several natural graph classes, but a weaker and meaningful notion—weak discrimination—admits a clean linear characterization in terms of the difference space induced by edge– type variations. This leads to general criteria based on support restrictions and linear independence of the weights over Q, covering forbidden edge types as well as other structural constraints. Finally, we investigate extremal values using the feasible polytope generated by edge–type vectors. Maximizers and minimizers of a VDB index correspond to normal cones of this polytope, providing a geometric explanation for the frequent appearance of stars and paths as extremal trees. Several explicit examples illustrate the framework.

Theenergy of a graph is a classical spectral invariant defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. Recently, the notion of local energy was introduced to measure the contribution of vertices to the total energy via vertex deletion. In this paper, we first study the variation of graph energy under the deletion of a set of vertices and obtain general upper bounds in terms of vertex degrees, together with a characterization of the equality cases under natural structural conditions. These results provide the foundation for extending the concept of local energy to digraphs. Using the singular values of the adjacency matrix, we define the local energy of a digraph and derive sharp upper bounds in terms of the in-degree and out-degree of a vertex. The equality cases are characterized by introducing a special class of vertices, called star-vertices. Finally, we obtain sharp bounds for the total local energy of a digraph in terms of its energy and of the Randić index.

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.