Date of Award

8-2011

Document Type

Thesis

Degree Name

Master of Science (MS)

College/School

College of Science and Mathematics

Department/Program

Mathematical Sciences

Abstract

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 >/ 2). The Randic and Sum Connectivity Index formulas are given for this tree (Chapter 5). This topic will be investigated more in future work.

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