Some Less Familiar Properties of Randić Index

Several mathematically relevant properties of the Randić connectivity index, that may be less familiar to the chemical community, are outlined and commented. This work is licensed under a Creative Commons Attribution 4.0 International License

Degree-Based Topological Indices

The degree of a vertex of a molecular graph is the number of first neighbors of this vertex. A large number of molecular-graph-based structure descriptors (topological indices) have been conceived, depending on vertex degrees. We summarize their main properties, and provide a critical comparative study thereof. (doi: 10.5562/cca2294

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

Applications of Multidimensional Space of Mathematical Molecular Descriptors in Large-Scale Bioactivity and Toxicity Prediction- Applications to Prediction of Mutagenicity and Blood-Brain Barrier Entry of Chemicals

In this chapter, we review our QSAR research in the prediction of toxicities, bioactivities and properties of chemicals using computed mathematical descriptors. Robust statistical methods have been used to develop high quality predictive quantitative structure-activity relationship (QSAR) models for the prediction of mutagenicity and BBB (blood-brain barrier) entry of two large and diverse sets chemicals. This work is licensed under a Creative Commons Attribution 4.0 International License

MIANN models of networks of biochemical reactions, ecosystems, and U.S. Supreme Court with Balaban-Markov indices

[Abstract] We can use Artificial Neural Networks (ANNs) and graph Topological Indices (TIs) to seek structure-property relationship. Balabans’ J index is one of the classic TIs for chemo-informatics studies. We used here Markov chains to generalize the J index and apply it to bioinformatics, systems biology, and social sciences. We seek new ANN models to show the discrimination power of the new indices at node level in three proof-of-concept experiments. First, we calculated more than 1,000,000 values of the new Balaban-Markov centralities Jk(i) and other indices for all nodes in >100 complex networks. In the three experiments, we found new MIANN models with >80% of Specificity (Sp) and Sensitivity (Sn) in train and validation series for Metabolic Reactions of Networks (MRNs) for 42 organisms (bacteria, yeast, nematode and plants), 73 Biological Interaction Webs or Networks (BINs), and 43 sub-networks of U.S. Supreme court citations in different decades from 1791 to 2005. This work may open a new route for the application of TIs to unravel hidden structure-property relationships in complex bio-molecular, ecological, and social networks

Distance-unbalancedness of graphs

In this paper we propose and study a new structural invariant for graphs, called distance-unbalanced\-ness, as a measure of how much a graph is (un)balanced in terms of distances. Explicit formulas are presented for several classes of well-known graphs. Distance-unbalancedness of trees is also studied. A few conjectures are stated and some open problems are proposed.Comment: 14 pages, 3 figure

Problèmes de réalisabilité et de connexité dans les graphes chimiques

RÉSUMÉ : Cette maîtrise s’inscrit dans le domaine de la chimie mathématique, et plus précisément le domaine de la théorie des graphes chimiques. Ce domaine est en pleine expansion depuis les travaux de William Cullen en 1758. Plusieurs invariants chimiques liés à la théorie des graphes ont été étudiés tels que l’indice de Randic, ainsi que le premier et le second indice de Zagreb. On les regroupe aujourd’hui sous le nom d’indices Adriatiques. Plusieurs modèles mathématiques ont été développés pour déterminer des bornes inférieures et supérieures pour ces invariants chimiques, ainsi que les graphes extrêmes correspondants. Ces modèles ne prennent pas en compte la connexité des graphes, en ce sens qu’aucune contrainte de connexité n’est imposée sur les graphes extrêmes générés. Nos recherches ont porté sur le développement d’un algorithme capable de générer des graphes extrêmes simples (c’est-à-dire sans boucle ni arête multiple) et connexes atteignant la kème plus petite ou la plus grande valeur des indices Adriatiques (k � 1). Pour ce faire, nous avons déterminé des conditions nécessaires et suffisantes pour l’existence d’un graphe simple et connexe lorsque pour toute paire i, j d’entiers strictement positifs, on impose le nombre mij d’arêtes reliant les sommets de degré i à ceux de degré j. Nous avons ensuite montré qu’il est possible d’imposer ces conditions à l’aide d’un modèle linéaire en nombres entiers. Puis, nous avons utilisé ce modèle mathématique pour déterminer les valeurs extrêmes de plusieurs indices Adriatiques pour plusieurs familles de graphes, et nous avons finalement montré comment il est possible de générer les graphes qui atteignent ces valeurs. Mots clés : graphes chimiques, invariants adriatiques, programmation linéaire en nombres entiers, problèmes de connexité.----------ABSTRACT : This research lies is in the field of mathematical chemistry, and more specifically in the area of the theory of chemical graphs. This domain is in full expansion since the early works of William Cullen in 1758. Several graphical chemical invariants have been studied such as the Randic index, and the first and second Zagreb index. These invariants are known today as the Adriatic invariants. Several mathematical models have been developed to determine lower and upper bounds for these chemical invariants, as well as their corresponding extreme graphs. However, these models do not take connexity constraints into account, in that sense that it is not imposed that the generated extreme graphs should be connected. The main objective of my research was to develop an algorithm able to generate extreme simple (i.e., without loops of multiple edges) and connected graphs that reach the kth smallest ou largest value of Adriatic invariants (k � 1). For this purpose, we have first determined necessary and sufficient conditions for the existence of a simple connected graph when for every pair i, j of striclty positive integers, it is imposed that the generated graphs should have a fixed number mij of edges linking vertices of degree i with vertices of degree j. We have then shown that these conditions can be imposed by an integer programming model. The model was then used to determine extremal values of Adriatic indices for several families of graphs, and we have finally shown how to generate graphs that reach these values. Key words: Chemical graphs, Adriatic indices, linear integer programming, connectivity problems

Computational and analytical studies of the Randic index in Erdös-Rényi models

In this work we perform computational and analytical studies of the Randic´ index R(G) in Erdös–Rényi models G(n, p) characterized by n vertices connected independently with probability p ∈ (0, 1). First, from a detailed scaling analysis, we show that R(G) = {R(G)}/(n/2) scales with the product ξ ≈ np, so we can define three regimes: a regime of mostly isolated vertices when ξ 10 (R(G) ≈ n/2). Then, motivated by the scaling of R(G), we analytically (i) obtain new relations connecting R(G) with other topological indices and characterize graphs which are extremal with respect to the relations obtained and (ii) apply these results in order to obtain inequalities on R(G) for graphs in Erdös–Rényi models.J.A.M.-B. acknowledges financial support from FAPESP (Grant No. 2019/06931-2), Brazil, and PRODEP-SEP (Grant No. 511- 6/2019.-11821), Mexico. J.M.R. and J.M.S. were supported in part by two grants from Ministerio de Economía y Competitividad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain

Modeling complex metabolic reactions, ecological systems, and financial and legal networks with MIANN models based on Markov-Wiener node descriptors

[Abstract] The use of numerical parameters in Complex Network analysis is expanding to new fields of application. At a molecular level, we can use them to describe the molecular structure of chemical entities, protein interactions, or metabolic networks. However, the applications are not restricted to the world of molecules and can be extended to the study of macroscopic nonliving systems, organisms, or even legal or social networks. On the other hand, the development of the field of Artificial Intelligence has led to the formulation of computational algorithms whose design is based on the structure and functioning of networks of biological neurons. These algorithms, called Artificial Neural Networks (ANNs), can be useful for the study of complex networks, since the numerical parameters that encode information of the network (for example centralities/node descriptors) can be used as inputs for the ANNs. The Wiener index (W) is a graph invariant widely used in chemoinformatics to quantify the molecular structure of drugs and to study complex networks. In this work, we explore for the first time the possibility of using Markov chains to calculate analogues of node distance numbers/W to describe complex networks from the point of view of their nodes. These parameters are called Markov-Wiener node descriptors of order kth (Wk). Please, note that these descriptors are not related to Markov-Wiener stochastic processes. Here, we calculated the Wk(i) values for a very high number of nodes (>100,000) in more than 100 different complex networks using the software MI-NODES. These networks were grouped according to the field of application. Molecular networks include the Metabolic Reaction Networks (MRNs) of 40 different organisms. In addition, we analyzed other biological and legal and social networks. These include the Interaction Web Database Biological Networks (IWDBNs), with 75 food webs or ecological systems and the Spanish Financial Law Network (SFLN). The calculated Wk(i) values were used as inputs for different ANNs in order to discriminate correct node connectivity patterns from incorrect random patterns. The MIANN models obtained present good values of Sensitivity/Specificity (%): MRNs (78/78), IWDBNs (90/88), and SFLN (86/84). These preliminary results are very promising from the point of view of a first exploratory study and suggest that the use of these models could be extended to the high-throughput re-evaluation of connectivity in known complex networks (collation)

WIENER INDEX OF SOME ACYCLIC GRAPHS

In the field of chemical graph theory, a topological index (a.k.a.connectivity index)is a type of molecular descriptor that is calculated based on the molecular graph of a chemical compound[5]. In this thesis, we have studied one of the well-known topological indices called Wiener. It is obtained by adding all the geodesic distances (or shortest paths) of the graph. As the number of vertices grows for anygraph, so does the Wiener number of that graph. We determine the Wiener values associated with several graphs, as functions of the number of vertices. We found that these infinite integer sequences have general formulae which include summations of triangular numbers. Further, we introduced new classes of trees and derived newinfinite integer sequences that are not available in the largest online encyclopedia of integer sequences