Abstract In this chapter we discuss differentiable structures on topological manifolds. particular, transversality, tubular neighbourhoods, index theory, the degree and theorems by Poincaré Hopf Bobylev Krasnosel’skiĭ.

Three new distance—based topological indices are described; two of them, D and D1 (mean distance topological indices, for any graphs, and for acyclic graphs, respectively) have a modest discriminating ability but may be useful for correlations, e. g. with octane numbers. The third index, J (average distance sum connectivity) is the least degenerate single topological index proposed till now. Th...

The Padmakar-Ivan (PI) index is a first-generation topological index (TI) based on sums over all edges between numbers of edges closer to one endpoint and numbers of edges closer to the other endpoint. Edges at equal distances from the two endpoints are ignored. An analogous definition is valid for the Wiener index W, with the difference that sums are replaced by products. A few other TIs are d...

Abstract In this study, we characterize the structure and some topological indices of a class random spider trees (RSTs) such as degree-based Gini index, Hoover generalized Zagreb other associated with these. We obtain exact asymptotic distributions number leaves via probabilistic methods. Moreover, relate model to RSTs that evolves in preferential attachment manner.

The eccentric connectivity index ξ is a novel distance–based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. It is defined as ξ(G) = ∑ v∈V (G) deg(v) · ε(v) , where deg(v) and ε(v) denote the vertex degree and eccentricity of v , respectively. We survey some mathematical properties of this index and furthermore support ...

The theoretical models based on valence connectivity index of the 3rd order, 3χv, have been discussed in terms of their ability to predict stability of coordination compounds. The key factors for the success are: (1) the choice of reliable experimental data for the calibration of the model, (2) writing an appropriate constitutional formula (i.e. graph) of the complex, and (3) development of pro...

The atom–bond connectivity (ABC) index of a graph G is defined as

the chromatic number of a graph g, denoted by χ(g), is the minimum number of colors such that g can be colored with these colors in such a way that no two adjacent vertices have the same color. a clique in a graph is a set of mutually adjacent vertices. the maximum size of a clique in a graph g is called the clique number of g. the turán graph tn(k) is a complete k-partite graph whose partition...

We introduce a partial order on the collection of chemical trees based on tree transformations. This partial order is tightly related to the Randić connectivity index χ. Its analysis provides new structural information about the behavior of χ. As an illustration of the approach presented, we give a different and more structural view of some known results about the first values of χ on the colle...