On some interconnections between combinatorial optimization and extremal graph theory

The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants under some constraints and with constructing extremal graphs. We analyze by examples some interconnections and interactions of the two theories and propose some conclusions

Algorithm and Complexity for a Network Assortativity Measure

We show that finding a graph realization with the minimum Randi\'c index for a given degree sequence is solvable in polynomial time by formulating the problem as a minimum weight perfect b-matching problem. However, the realization found via this reduction is not guaranteed to be connected. Approximating the minimum weight b-matching problem subject to a connectivity constraint is shown to be NP-Hard. For instances in which the optimal solution to the minimum Randi\'c index problem is not connected, we describe a heuristic to connect the graph using pairwise edge exchanges that preserves the degree sequence. In our computational experiments, the heuristic performs well and the Randi\'c index of the realization after our heuristic is within 3% of the unconstrained optimal value on average. Although we focus on minimizing the Randi\'c index, our results extend to maximizing the Randi\'c index as well. Applications of the Randi\'c index to synchronization of neuronal networks controlling respiration in mammals and to normalizing cortical thickness networks in diagnosing individuals with dementia are provided.Comment: Added additional section on application

Randic and Sum Connectivity Indices of Certain Trees

The focus of this thesis is on new development of the Randic and Sum Connec­tivity Indices of certain molecular and symmetric trees representing acyclic alkanes, or aliphatic hydrocarbons. The Randic Connectivity Index is one of the most used molecular descriptors in Quantitative Structure-Property and Structure-Activity Relationship modeling because of the relation that the isomers have between their properties and their structure. The structure-boiling point relationship models of aliphatic alcohols have been studied using the Sum Connectivity Index and compared to the Randic Connectivity Index. A specific type of tree Tn,a, well-known in graph theory as a double star, was studied by Zhou and Trinajstic. In this thesis, Tn,a trees are investigated. The tree Tn,2 which has the third smallest Sum Connectivity Index value among all the trees with n vertices is found to be interesting and thereby is further explored. Some alkane trees are symmetric, which is the concentration of this thesis. The symmetric double star trees are denoted by Jn· The tree Jn has n vertices and is built on the path P₂ with (n - 2)/2 leaves from each vertex of the path. The Randic and Sum Connectivity Index formulas of the symmetric tree ln are developed. Also, estimations of the Randic and Sum Connectivity Indices of Jn are given. Relationships and comparisons between the Randic and Sum Connectivity Indices are analyzed in respect to the tree Jn· The ratio and difference of the Randic and Sum Connectivity Indices are further discussed. The thesis starts with the history of the indices of molecular trees in Chemistry and Biology (Chapter 1). Chapter 2 provides a list of observations of the properties of both connectivity indices of the related trees. The symmetric tree Jn is discussed in Chapter 3, in which formulas and properties of the Randic and Sum Connectivity Indices are given. The main results of the thesis are reported in Chapter 4, where the graphs which have the maximal or minimal Randic and Sum Connectivity values among all Tn,a graphs with n vertices are identified. The closeness of the two indices of Tn,a trees is also discussed. The paper concludes with a similar tree, denoted Tn,a x Pm, extended from the tree Tn,a by replacing the middle path P2 with the path Pm (m \u3e/ 2). The Randic and Sum Connectivity Index formulas are given for this tree (Chapter 5). This topic will be investigated more in future work

Variable neighborhood search for extremal graphs. 6. Analyzing bounds for the connectivity index

Recently, Araujo and De la Peña gave bounds for the connectivity index of chemical trees as a function of this index for general trees and the ramification index of trees. They also gave bounds for the connectivity index of chemical graphs as a function of this index for maximal subgraphs which are trees and the cyclomatic number of the graphs. The ramification index of a tree is first shown to be equal to the number of pending vertices minus 2. Then, in view of extremal graphs obtained with the system AutoGraphiX, all bounds of Araujo and De la Peña are improved, yielding tight bounds, and in one case corrected. Moreover, chemical trees of a given order and a number of pending vertices with minimum and with maximum connectivity index are characterized.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Variable neighborhood search for extremal graphs. 6. Analyzing bounds for the connectivity index

Recently, Araujo and De la Peña gave bounds for the connectivity index of chemical trees as a function of this index for general trees and the ramification index of trees. They also gave bounds for the connectivity index of chemical graphs as a function of this index for maximal subgraphs which are trees and the cyclomatic number of the graphs. The ramification index of a tree is first shown to be equal to the number of pending vertices minus 2. Then, in view of extremal graphs obtained with the system AutoGraphiX, all bounds of Araujo and De la Peña are improved, yielding tight bounds, and in one case corrected. Moreover, chemical trees of a given order and a number of pending vertices with minimum and with maximum connectivity index are characterized.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Invariantes espetrais da matriz de Randic de um grafo

Este trabalho apresenta um estudo sobre invariantes espetrais da matriz de Randić de um grafo. O índice de Randić é um invariante espetral apresentado em 1975 por Milan Randić e com importantes aplicações ao nível da Química. Em 2010 define-se a matriz de Randić, uma matriz não negativa construída a partir desse índice. O estudo do espetro de matrizes associadas a grafos é um dos grandes objetivos da investigação em teoria dos grafos e são já diversas as aplicações em diferentes áreas científicas. Neste trabalho é estudado o espetro da matriz de Randić associada a um grafo e definido o spread de Randić. Para além disso, são apresentados majorantes e minorantes para esse invariante espetral. Em Química, a energia de grafos caterpillar, que estão associados a sistemas aromáticos, está relacionada com as relações de ressonância desses sistemas. Tendo esse facto como motivação, é estudado o espetro e o espetro de Randić de grafos caterpillar e são apresentados majorantes para a energia e para a energia de Randić dessa classe de grafos.This work presents a study related to spectral invariants for the Randić matrix of a graph. The Randić index is a spectral invariant presented in 1975 by Milan Randić and with important applications in chemistry. In 2010 the Randić matrix was defined as a nonnegative matrix built from this index. The study of the spectrum of matrices associated with graphs is one of the major goals of research in graph theory and there are already several applications in different scientific areas. In this work the spectrum of the Randić matrix associated to a graph is studied and the Randić spread is defined. In addition, upper and lower bounds are presented for this spectral invariant. In chemistry, the energy of caterpillar graphs, that are associated with aromatic systems, is related with the resonance of these systems. Having this as motivation, the spectrum and the Randić spectrum of caterpillar graphs are studied and upper bounds are presented for the energy and for the Randić energy of this class of graphs.Programa Doutoral em Matemátic

The maximum Randic index of chemical trees with k pendants

GRF, Research Grant Council of Hong Kong; FRG, Hong Kong Baptist University; National Natural Science Foundation of China [10831001]A tree is a chemical tree if its maximum degree is at most 4. Hansen and Melot [P. Hansen, H. Melot, Variable neighborhood search for extremal graphs 6: analyzing bounds for the connectivity index, J. Chem. Inf Comput. Sci. 43 (2003) 1-14], Li and Shi [X. Li, Y.T. Shi, Corrections of proofs for Hansen and Melot's two theorems, Discrete Appl. Math., 155 (2007) 2365-2370] investigated extremal Randic indices of the chemical trees of order n with k pendants. In their papers, they obtained that an upper bound for Randic index is n/2 + (3 root 2+root 6-7)k/6. This upper bound is sharp for n >= 3k - 2 but not for n < 3k - 2. In this paper, we find the maximum Randic index for n < 3k - 2. Examples of chemical trees corresponding to the maximum Randic indices are also constructed. (C) 2009 Elsevier B.V. All rights reserved