A proof for a conjecture on the Randić index of graphs with diameter

AbstractThe Randić index R(G) of a graph G is defined by R(G)=∑uv1d(u)d(v), where d(u) is the degree of a vertex u in G and the summation extends over all edges uv of G. Aouchiche et al. proposed a conjecture on the relationship between the Randić index and the diameter: for any connected graph on n≥3 vertices with the Randić index R(G) and the diameter D(G), R(G)−D(G)≥2−n+12andR(G)D(G)≥n−3+222n−2, with equalities if and only if G is a path. In this work, we show that this conjecture is true for trees. Furthermore, we prove that for any connected graph on n≥3 vertices with the Randić index R(G) and the diameter D(G), R(G)−D(G)≥2−n+12, with equality if and only if G is a path

GTI-space : the space of generalized topological indices

A new extension of the generalized topological indices (GTI) approach is carried out torepresent 'simple' and 'composite' topological indices (TIs) in an unified way. Thisapproach defines a GTI-space from which both simple and composite TIs represent particular subspaces. Accordingly, simple TIs such as Wiener, Balaban, Zagreb, Harary and Randićconnectivity indices are expressed by means of the same GTI representation introduced for composite TIs such as hyper-Wiener, molecular topological index (MTI), Gutman index andreverse MTI. Using GTI-space approach we easily identify mathematical relations between some composite and simple indices, such as the relationship between hyper-Wiener and Wiener index and the relation between MTI and first Zagreb index. The relation of the GTI space with the sub-structural cluster expansion of property/activity is also analysed and some routes for the applications of this approach to QSPR/QSAR are also given

On the Randić index and girth of graphs

AbstractThe Randić index R(G) of a graph G is defined by R(G)=∑uv1d(u)d(v), where d(u) is the degree of a vertex u in G and the summation extends over all edges uv of G. In this work, we give a sharp upper bound and a lower bound of the Randić index among connected n-vertex graphs with girth g≥k(k≥3)

Bond Additive Modeling 1. Adriatic Indices

Some of the most famous molecular descriptors are bond additive, i.e. they are calculated as the sum of edge contributions (Randić-type indices, Balaban-type indices, Wiener index and its modifications, Szeged index...). In this paper, the methods of calculations of bond contributions of these descriptors are analyzed. The general concepts are extracted, and based on these concepts a large class of molecular descriptors is defined. These descriptors are named Adriatic indices. An especially interesting subclass of these descriptors consists of 148 discrete Adriatic indices. They are analyzed on the testing sets provided by the International Academy of Mathematical Chemistry, and it has been shown that they have good predictive properties in many cases. They can be easily encoded in the computer and it may be of interest to incorporate them in the existing software packages for chemical modeling. It is possible that they could improve various QSAR and QSPR studies

Molecular Topology. 14. Molord Algorithm and Real Number Subgraph Invariants

An algorithm, MOLORD, is proposed for defining real number invariants for subgraphs of various sizes in molecular graphs. The algorithm is based on iterative line derivati ves and accounts for heteroatoms by means of their electronegativities. It can be used in topological equivalence perception as well as to provide local and global descriptors for QSPR or QSAR studies. The algorithm is implemented on a TURBO PASCAL, TOPIND program and examplified on a set of selected graphs

A Vibrational Approach to Node Centrality and Vulnerability in Complex Networks

We propose a new measure of vulnerability of a node in a complex network. The measure is based on the analogy in which the nodes of the network are represented by balls and the links are identified with springs. We define the measure as the node displacement, or the amplitude of vibration of each node, under fluctuation due to the thermal bath in which the network is supposed to be submerged. We prove exact relations among the thus defined node displacement, the information centrality and the Kirchhoff index. The relation between the first two suggests that the node displacement has a better resolution of the vulnerability than the information centrality, because the latter is the sum of the local node displacement and the node displacement averaged over the entire network.Comment: 27 page

The Rücker–Markov invariants of complex bio-systems: applications in parasitology and neuroinformatics

[Abstract] Rücker's walk count (WC) indices are well-known topological indices (TIs) used in Chemoinformatics to quantify the molecular structure of drugs represented by a graph in Quantitative structure–activity/property relationship (QSAR/QSPR) studies. In this work, we introduce for the first time the higher-order (kth order) analogues (WCk) of these indices using Markov chains. In addition, we report new QSPR models for large complex networks of different Bio-Systems useful in Parasitology and Neuroinformatics. The new type of QSPR models can be used for model checking to calculate numerical scores S(Lij) for links Lij (checking or re-evaluation of network connectivity) in large networks of all these fields. The method may be summarized as follows: (i) first, the WCk(j) values are calculated for all jth nodes in a complex network already created; (ii) A linear discriminant analysis (LDA) is used to seek a linear equation that discriminates connected or linked (Lij = 1) pairs of nodes experimentally confirmed from non-linked ones (Lij = 0); (iii) The new model is validated with external series of pairs of nodes; (iv) The equation obtained is used to re-evaluate the connectivity quality of the network, connecting/disconnecting nodes based on the quality scores calculated with the new connectivity function. The linear QSPR models obtained yielded the following results in terms of overall test accuracy for re-construction of complex networks of different Bio-Systems: parasite–host networks (93.14%), NW Spain fasciolosis spreading networks (71.42/70.18%) and CoCoMac Brain Cortex co-activation network (86.40%). Thus, this work can contribute to the computational re-evaluation or model checking of connectivity (collation) in complex systems of any science field.Programa Iberoamericano de Ciencia y Tecnología para el Desarrollo; Ibero-NBIC, 209RT-0366Ministerio de Ciencia e Innovación; TIN2009-0770

Inequalities on Topological Indices

Topological indices have been widely used in different fields associated with scientific research. They are recognized as useful tools in applied research in Chemistry, Ecology, Biology, Physics, among others. For many years, scientists have been trying to improve the predictive power of the famous Randi’c index. This led to the introduction and study of new topological descriptors that correlate or improve the level of prediction of the Randi’c index. Among the most commonly used descriptors are the Inverse index, the first general Zagreb index and the recently introduced Arithmetic- Geometric index. In this work we study the mathematical properties and relationships of the aforementioned topological indices.Programa de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidente: Domingo de Guzmán Pestana Galván.- Secretaria: Ana Portilla Ferreira.- Vocal: Eva Tourís Loj