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- W101771870 abstract "The number of independent subsets and the energy of trees E. O. D. Andriantiana Department of Mathematics, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa. Thesis: MSc September 2010 The Merri eld-Simmons index σ, de ned as the total number of independent vertex subsets, the Hosoya index Z, de ned similarly as the total number of independent edge subsets, and the energy En of a graph, which is the sum of the absolute values of its eigenvalues, are among the most popular parameters studied in chemical graph theory. For molecular graphs (in particular trees), they are known to be correlated to some physico-chemical properties of the corresponding compounds. In this thesis, we rst introduce the three parameters together with some of their basic properties. Then we discuss certain techniques to characterize the trees for which the maximum/minimum of σ, Z and En are reached. We also study several special classes of trees, such as trees with prescribed diameter or maximum degree, and provide additional results on trees with large Hosoya index or small Merri eld-Simmons index. Finally, in the main part of this thesis, we study the three graph invariants for the class T1,d of trees whose vertex degrees are restricted to either 1 or d (for some d ≥ 3), which is a natural restriction in the chemical context. We nd that the minimum of the Merri eld-Simmons index and the maximum of the Hosoya index are both attained for path-like trees which is followed by sequences of what we call generalized tripods. Other types of trees only appear much later in a list sorted with respect to the two graph invariants σ and Z. Elements of T1,d with maximum Merri eld-Simmons index and with minimum Hosoya index are described as well, and similar results are also found for the energy. Comparing the behaviours of the three parameters for the set of all trees without restriction and the set T1,d shows some interesting analogies." @default.
- W101771870 created "2016-06-24" @default.
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- W101771870 date "2010-12-01" @default.
- W101771870 modified "2023-09-23" @default.
- W101771870 title "The number of independent subsets and the energy of trees" @default.
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