The number of independent subsets and the energy of caterpillars under degree restriction
- Authors: Xhanti, Sinoxolo
- Date: 2024-10-11
- Subjects: Graph energy , Independent set , Hosoya index , Chemical graph theory , Caterpillar tree , Graph theory
- Language: English
- Type: Academic theses , Doctoral theses , text
- Identifier: http://hdl.handle.net/10962/466838 , vital:76791 , DOI https://doi.org/10.21504/10962/466838
- Description: The energy En(G) of a graph G is defined as the sum of the absolute values of its eigenvalues. The Hosoya index Z(G) of a graph G is the number of independent edge subsets of G, including the empty set. And, the Merrifield-Simmons index σ(G) of a graph G is the number of independent vertex subsets of G, including the empty set. The studies of these three graph invariants are motivated by their application in chemistry, combined with pure mathematical interests. In particular, they can be used to predict boiling points of saturated hydrocarbons and estimate the total π-electron energy. For ℓ ≥ 1, let a1, a2, . . . , aℓ be non-negative integers, such that a1 and aℓ are positive. The tree obtained from the path graph of vertices v1, v2, . . . , vℓ, by attaching ai new leaves to vi, for 1 ≤ i ≤ ℓ, is called a (a1, a2, . . . , aℓ)-caterpillar and denoted by C(a1 + 1, a2 + 2, . . . , aℓ−1 + 2, aℓ +1). In this thesis, we characterize extremal caterpillars relative to the energy, the Hosoya index and the Merrifield-Simmons index. We first study caterpillars with the same degree sequence, then compare caterpillars of the same size, same order, and different degree sequence. For any given degree sequence D, we characterize the caterpillar X(D) that maximizes Z and En. In X(D), as we move along the internal path towards the center, the degrees are in a nondecreasing order. Characterization of the caterpillar S(D) that has the minimum Z and En and maximum σ is also provided. In S(D), large and small degrees alternate. , Thesis (PhD) -- Faculty of Science, Mathematics, 2024
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- Date Issued: 2024-10-11
On the Wiener index of bicyclic graphs and graphs with fixed segment sequence
- Authors: Xhanti, Sinoxolo
- Date: 2021-10-29
- Subjects: Graph theory , Chemistry Mathematics , Chemistry Graphic methods , Wiener index , Bicyclic graphs , Fixed segment sequence , Degree sequence , Circumference , Core
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/190700 , vital:45019
- Description: Wiener index is defined as the sum of the distances between all unordered pairs of vertices in a graph. The study of the Wiener index is motivated by its application in chemistry. This thesis focuses on finding extremal bicyclic graphs relative to Wiener index under various conditions such as fixed circumference (length of the longest cycle) or fixed size of the core (maximal subgraph with no degree less than 2). A segment of a graph G is either a path whose end vertices have degree 1 or at least 3 in G and all the internal vertices have degree 2 in G, or a cycle where all the vertices have degree 2 in G except possibly one. The lengths of all the segments of G form it segment sequence. We also discuss extremal graphs with given segment sequence. , Thesis (MSc) -- Faculty of Science, Mathematics, 2021
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- Date Issued: 2021-10-29
A method for the evaluation of similarity measures on graphs and network-structured data
- Authors: Naude, Kevin Alexander
- Date: 2014
- Subjects: Graph theory , Network analysis (Planning)
- Language: English
- Type: Thesis , Doctoral , DPhil
- Identifier: vital:10494
- Description: Measures of similarity play a subtle but important role in a large number of disciplines. For example, a researcher in bioinformatics may devise a new computed measure of similarity between biological structures, and use its scores to infer bio-logical association. Other academics may use related approaches in structured text search, or for object recognition in computer vision. These are diverse and practical applications of similarity. A critical question is this: to what extent can a given similarity measure be trusted? This is a difficult problem, at the heart of which lies the broader issue: what exactly constitutes good similarity judgement? This research presents the view that similarity measures have properties of judgement that are intrinsic to their formulation, and that such properties are measurable. The problem of comparing similarity measures is one of identifying ground-truths for similarity. The approach taken in this work is to examine the relative ordering of graph pairs, when compared with respect to a common reference graph. Ground- truth outcomes are obtained from a novel theory: the theory of irreducible change in graphs. This theory supports stronger claims than those made for edit distances. Whereas edit distances are sensitive to a configuration of costs, irreducible change under the new theory is independent of such parameters. Ground-truth data is obtained by isolating test cases for which a common outcome is assured for all possible least measures of change that can be formulated within a chosen change descriptor space. By isolating these specific cases, and excluding others, the research introduces a framework for evaluating similarity measures on mathematically defensible grounds. The evaluation method is demonstrated in a series of case studies which evaluate the similarity performance of known graph similarity measures. The findings of these experiments provide the first general characterisation of common similarity measures over a wide range of graph properties. The similarity computed from the maximum common induced subgraph (Dice-MCIS) is shown to provide good general similarity judgement. However, it is shown that Blondel's similarity measure can exceed the judgement sensitivity of Dice-MCIS, provided the graphs have both sufficient attribute label diversity, and edge density. The final contribution is the introduction of a new similarity measure for graphs, which is shown to have statistically greater judgement sensitivity than all other measures examined. All of these findings are made possible through the theory of irreducible change in graphs. The research provides the first mathematical basis for reasoning about the quality of similarity judgments. This enables researchers to analyse similarity measures directly, making similarity measures first class objects of scientific inquiry.
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- Date Issued: 2014