New Trends in Majorization Techniques for Bounding Topological Indices

The appeal of the research on molecular descriptors in Mathematical Chemistry stems, in no small measure, from the wide variety of approaches to the subject and their fruitful interactions. In the paragraphs that follow we will focus on one of those viewpoints, the majorization technique, with some support from the theory of electric networks. Majorization allows to find upper and lower bounds for many descriptors through the identification of maximal and minimal tuples in some subsets of the $n$-dimensional real space, endowed with a partial order, and the monotonicity of the descriptors when thought of as Schur-convex functions defined on those suitable subspaces. On the other hand, the theory of electric networks provides a number of sum rules for the effective resistances, such as Foster's theorems, and also a monotonicity notion, Rayleigh's monotonicity principle. These electric ideas are applicable not only to the resistive descriptors, such as the Kirchhoff index and its relatives, but also to other descriptors defined in terms only of the degrees of the vertices, such as the ABC and AZI indices. In these cases, the majorization is performed on the effective resistances, not on the degrees of the vertices or on the eigenvalues of the graph. In general, proofs are omitted for the sake of brevity, although these can be found in the references provided. A final section on numerical results compares in some specific instances our results obtained with majorization and those found in the literature.